Sigmoid Functions and Explanations

Sigmoid function are also common in statistics as cumula cumulative tive distribution functions (which functions  (which go from 0 to 1), such as the integrals of the logistic the  logistic distribution, distribution, the normal the  normal distribution, distribution, and Student’s t  probability density functions. functions.

1

0.5

1

A sigmoid function is a bounded differentiable real function that is defined for all real input values and has a nonnegative derivative at each point.[1]

0

−6

−4

−2

0

2

4

Defin Definiitio tion

6

The logistic The logistic curve

2 1.00

Pro Propert pertiies

In general, a sigmoid function is real is  real-valued, -valued, monotonic, and differentiable and  differentiable   having a   non-negative  non-negative   first  derivative which is bell shaped. A sigmoid function function is constrained  x by a pair of horizontal of  horizontal asymptotes as asymptotes  as x .

0.75

0.50

→ ±∞

0.25

       )      x        (

        f0.00       r       e

−0.25

3

−0.50

Exam Examp ples

−0.75 −1.00 −3

− 2

−1

0

1

2

3

x

Plot of the error the error function

A sigmoid function  is a  mathematical function having function having a characteristic “S"-shaped curve or  sigmoid curve . Often,   sigmoid function function   refers to the special case of the logistic function shown function  shown in the first figure and defined by the formula

S (x) =

Some sigmoid functions compared. In the drawing all functions  are normalized in such a way that their slope at the origin is 1.

1 . 1 + e−x

•   Logistic function

Other examples of similar shapes include the Gompertz the  Gompertz curve   (used in modeling systems that saturate at large values of x) and the ogee the  ogee curve (used curve  (used in the spillway the  spillway of  of some dams some  dams). ). Sigmoid functions have have domain of all real numbers, numbers, with return value monotonically value monotonically increasing most often from 0 to 1 or alternatively from −1 to 1, depending on convention.

f (x) =

1 1 + e−x

•   hyperbolic tangent

A wide variety of sigmoid functions have been used as the activation the activation function of function of artificial  artificial neurons, neurons, including the logisti logisticc and hyperbo hyperbolic lic tangent func functions. tions. Sigmoid curves

ex e−x f (x) =  tanh x  = x e + e−x

−

1

2

5

•   arctangent function

•   Cumulative distribution function •   Generalized logistic curve •   Gompertz function •   Heaviside step function •   Hyperbolic function •  Logistic distribution •   Logistic function •   Logistic regression •   Logit •  Modified hyperbolic tangent •   Softplus function •   Smoothstep function Smoothstep function (Graphics) •   Softmax function •   Weibull distribution •   Netoid function

f (x) =  arctan x

•   Gudermannian function x

� 

f (x) =  gd (x) =

0

1 cosh t

 dt

•   Error function f (x) =  erf (x) =

2

√ π

x

� 

2

e−t dt

0

•   Generalised logistic function f (x) = (1 + e−x )−α ,

α >  0

5

•   Smoothstep function Smoothstep function

�∫  � � � ∫  � � 1−u 1−u du f (x) = sgn(x) 1

2

N 

−1

0

x

0

2

N 

du

•   Specific algebraic Specific  algebraic functions f (x) =

2

The   integral   of any continuo continuous, us, non-n non-nega egativ tive, e, “bump“bumpshaped” function will be sigmoidal, thus the  cumulative distribution functions for functions for many common probability common probability distributions   are sigmoidal. tributions sigmoidal. One such example example is the error function, which is related to the cumulative the  cumulative distribution function (CDF) of (CDF) of a  normal distributio distribution n. Many natural processes, such as t hose of complex complex system learning curves, curves, exhibit a progression from small beginnings that accelerates accelerates and approaches a climax over over time. When When a speci specific fic mathe mathema matic tical al model model is lackin lacking, g, a sigm sigmoi oid d [2] function func tion is often used.

4

Ref Referen erence cess

[1] Han, Jun; Morag, Morag, Claudio Claudio (1995). (1995). “The influence influence of the sigmoid function parameters on the speed of backpropagatio agation n learning”. learning”. In Mira, José; Sandov Sandoval, al, Francisco. Francisco. 1 x . pp. 195– From Natural to Artificial Neural Computation N  1 201. 1 x

| |≤ | |≥

 ≥  ≥

[2] Gibbs, M.N. (Nov 2000). 2000). “Variational “Variational Gaussian Gaussian process process classifiers”.   IEEE Transactions on Neural Networks . 11 (6): 1458–1464. doi 1458–1464.  doi::10.1109/72.883477 10.1109/72.883477..

•   Mitch Mitchel ell, l,

x

√ 1 + x

REFE REFERE RENC NCES  ES 

See also

•  Activation function

Tom Tom M. (1997) (1997).. Machine Learning. WCB–McGraw–Hill.   ISBN 0-07-042807-7. 0-07-042807-7.. In particul particular ar see “Chapter “Chapter 4: Artificia Artificiall Neural Networks” works” (in particular particular pp. 96–97) 96–97) where where Mitchell Mitchell uses the word “logistic function” and the “sigmoid function” synonymously – this function he also calls the “squashing function” – and the sigmoid (aka logistic) function is used to compress the outputs of the “neurons” in multi-layer neural nets.

•  Humphrys, Mark.  “Continuous output, the sigmoid function”. Properties of the sigmoid, including how function”. it can shift along axes and how its domain may be transformed.

2

5

•   arctangent function

•   Cumulative distribution function •   Generalized logistic curve •   Gompertz function •   Heaviside step function •   Hyperbolic function •  Logistic distribution •   Logistic function •   Logistic regression •   Logit •  Modified hyperbolic tangent •   Softplus function •   Smoothstep function Smoothstep function (Graphics) •   Softmax function •   Weibull distribution •   Netoid function

f (x) =  arctan x

•   Gudermannian function x

� 

f (x) =  gd (x) =

0

1 cosh t

 dt

•   Error function f (x) =  erf (x) =

2

√ π

x

� 

2

e−t dt

0

•   Generalised logistic function f (x) = (1 + e−x )−α ,

α >  0

5

•   Smoothstep function Smoothstep function

�∫  � � � ∫  � � 1−u 1−u du f (x) = sgn(x) 1

2

N 

−1

0

x

0

2

N 

du

•   Specific algebraic Specific  algebraic functions f (x) =

2

The   integral   of any continuo continuous, us, non-n non-nega egativ tive, e, “bump“bumpshaped” function will be sigmoidal, thus the  cumulative distribution functions for functions for many common probability common probability distributions   are sigmoidal. tributions sigmoidal. One such example example is the error function, which is related to the cumulative the  cumulative distribution function (CDF) of (CDF) of a  normal distributio distribution n. Many natural processes, such as t hose of complex complex system learning curves, curves, exhibit a progression from small beginnings that accelerates accelerates and approaches a climax over over time. When When a speci specific fic mathe mathema matic tical al model model is lackin lacking, g, a sigm sigmoi oid d [2] function func tion is often used.

4

Ref Referen erence cess

[1] Han, Jun; Morag, Morag, Claudio Claudio (1995). (1995). “The influence influence of the sigmoid function parameters on the speed of backpropagatio agation n learning”. learning”. In Mira, José; Sandov Sandoval, al, Francisco. Francisco. 1 x . pp. 195– From Natural to Artificial Neural Computation N  1 201. 1 x

| |≤ | |≥

 ≥  ≥

[2] Gibbs, M.N. (Nov 2000). 2000). “Variational “Variational Gaussian Gaussian process process classifiers”.   IEEE Transactions on Neural Networks . 11 (6): 1458–1464. doi 1458–1464.  doi::10.1109/72.883477 10.1109/72.883477..

•   Mitch Mitchel ell, l,

x

√ 1 + x

REFE REFERE RENC NCES  ES 

See also

•  Activation function

Tom Tom M. (1997) (1997).. Machine Learning. WCB–McGraw–Hill.   ISBN 0-07-042807-7. 0-07-042807-7.. In particul particular ar see “Chapter “Chapter 4: Artificia Artificiall Neural Networks” works” (in particular particular pp. 96–97) 96–97) where where Mitchell Mitchell uses the word “logistic function” and the “sigmoid function” synonymously – this function he also calls the “squashing function” – and the sigmoid (aka logistic) function is used to compress the outputs of the “neurons” in multi-layer neural nets.

•  Humphrys, Mark.  “Continuous output, the sigmoid function”. Properties of the sigmoid, including how function”. it can shift along axes and how its domain may be transformed.

3

6

Text Text and and image image sources, sources, contrib contributors, utors, and lic license ensess

6.1 6.1 •

6.2 6.2 •

•

•

•

•

Text   Sigmoid function  Source:  https://en.wikipedia.org/wiki/Sigmoid_function?oldid=781040006 Contributors:  The   The Anome, Michael Hardy, Chris-martin, Kku, Chinju, Glenn, AnthonyQBachler, AnthonyQBachler, Pmcray, Tea2min, Giftlite, BenFrantzDale, Chinasaur, Jorge Stolfi, Mboverload, Eequor, MarkSweep, Cacycle, Bender235, ZeroOne, Wolfman, Linas, Jacobolus, Doubleddoubleu, GregorB, Waldir, Wdanwatts, Melesse, Strait, HappyCamper, Kri, R oboto de Ajvol, YurikBot, Cleared as filed, Daniel Mietchen, Light current, Closedmouth, Cmglee, TravisTX, SmackBot, Jfmiller28, CapitalSasha, KennethJ, KennethJ, ZackV, Nbarth, Michael.Pohoreski, Michael.Pohoreski, Georg-Johann, Georg-Johann, Wvbailey, Dr. Sunglasses, Sbmehta, Freewol, Tktktk, Jim.belk, JHunterJ, Knights who say ni, Hyperwiz, A. Pichler, Trialsanderrors, Trialsanderrors, Pfhenshaw, Pfhenshaw, Schroding79, Schroding79, Thijs!bot, Thijs!bot, Mkch, Wasell, Paul Haymon, Pebkac, Loluengo, Daniel5Ko, Akeron, Barraki, Red Act, Technopat, Hagman, SieBot, Thelostchild, ShadowPhox, Mike2vil, Dolphin51, Addbot, MrOllie, New Image Uploader 929, Cesiumfrog, Alfie66, Luckas-bot, AnomieBOT, Csigabi, Materialscientist, LilHelpa, Xqbot, Isheden, Sławomir Biały, John85, Tom.Reding, Vjost, Inferior Olive, Helwr, EmausBot, Slawekb, ZéroBot, Mrmatiko, Glosser.ca, ClueBot NG, KlappCK, Marcocapelle, Manish Singh321, ChrisGualtieri, Elen so, Cerabot~enwiki, Defpo, Monkbot, Lolo often, ZxzStar, Deacon Vorbis and Anonymous: 73

Imag Images es   File:Commons-logo.svg  Source:   https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License:  PD  Contributors:  ?  ?  Origi ? nal artist:  ?   File:Error_Function.svg  Source:    https://upload.wikimedia.org/wikipedia/commons/2/2f/Error_Function.svg   License:    Public domain Contributors:  self-made,   self-made, Inkscape  Original artist:   Inductiveload   File:Gjl-t(x).svg   Source:    https://upload.wikimedia.org/wikipedia/commons/6/6f/Gjl-t%28x%29.svg   License:  CC BY-SA BY-SA 3.0   Contributors:    This   vector vector image   inclu includes des eleme elements nts that that have have been been taken taken or adapte adapted d from this: this:   Error func function.svg tion.svg (by Geek3). Geek3 ).  Original artist:  Georg-Johann   File:Logistic-curve.svg Source:  https://upload.wikimedia.org/wikipedia/commons/8/88/Logistic-curve.svg License:  Public domain  Contributors:  Created  Created from scratch with gnuplot  Original artist:  Qef  (talk  ( talk))   File:Question_book-new.svg  Source:    https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg  License:    Cc-by-sa-3.0 Contributors:  Created from scratch in Adobe Illustrator. Based on Image:Question on  Image:Question book.png created book.png  created by User:Equazcion by  User:Equazcion  Original artist:  Tkgd2007

6.3 •

Conte Content nt lice license nse Creativee Commons Attribution-Share Creativ Attribution-Share Alike 3.0

Logistic function 1

For the recurrence relation, see Logistic see  Logistic map. map. A  logistic function  or  logistic curve  is a common “S”

Mathe Mathema mati tical cal proper properti ties es

The standard The standard logistic function function is  is the logistic function with parameters (k  (k  =  = 1, x  1,  x 0  = 0, L 0,  L =  = 1) which yields

1

f (x) = 0.5

1 . 1 +  e−x

In practice, due to the nature of the exponential the  exponential function − x  e , it is often sufficient to compute the standard logistic function for x  for x  over  over a small range of real numbers such as a range contained in [−6, +6]. 0

−6

−4

−2

0

2

4

6

1.1 Standard logistic sigmoid function i.e. L  = 1, k

Deriv Derivat ativ ivee

= 1, x0 = 0

The standard logistic function has an easily calculated derivative:

shape (sigmoid (sigmoid curve), curve), with equation:

f (x) = f (x) =

L

1 + e−k(x−x

0

)

where

1 1+e−

x

d f (x) dx

=

d f (x) dx

=

x

e

e

=

x

1+e

x

(1+e )−e ·e (1+e )2 x

·

x

x

x

e

x

(1+e )2 x

=  f (x)(1 −  f (x))

The logistic function also has the property that:

the natural logarithm base base (also (also known known as Euler’s • e = the number), number ),

1 −  f (x) =  f (−x).

the  x -value -value of the sigmoid’s midpoint, • x 0  = the x 

Thus, x Thus, x  →  f (x)  −  1/2  is an odd an  odd function. function.

• L  = the curve’s maximum value, and

The derivative of the logistic function has the property that:

 = the steepness of the curve. [1] • k  = d d f (x) = f (−x). dx dx

For values of x   x   in the range of  real numbers  from −∞ to +∞, the S-curve shown on the right is obtained (with the graph of f   f   approaching L as x   x   approaches +∞ and approaching approaching zero as x  as  x  approaches   approaches −∞).

1.2

The The funct unctio ion n was was name named d in 1844 1844–1 –184 845 5 by   Pierre François Ver Verhulst hulst,, who who studie studied d it in relat relatio ion n to popula populatio tion n [2] growth. The initial stage of growth is approximately exponential;; then, as saturation begins, the growth slows, exponential and at maturity, growth stops.

Logisti Logisticc diff differen erential tial equatio equation n

The standa standard rd logi logisti sticc func unctio tion n is the soluti solution on of the simpl simplee first-order non-linear ordinary non-linear ordinary differential equation

d f (x) =  f (x)(1  −  f (x)) dx

The logisti logisticc functi function on finds finds applic applicatio ations ns in a range range of fields fields,, includi including ng   artificia artificiall neu neural ral netw network orkss,   biology (especially   ecology), ecology),   biomathematics, biomathematics,   chemistry, chemistry, demography,,   economics, demography economics,   geoscience, geoscience,   mathematical psychology,,   probability, psychology probability,   sociology, sociology,   political political scienc sciencee, linguistics,, and statistics linguistics and  statistics..

with boundary with boundary conditio condition n f (0)  f (0) = 1/2. This equation is the continuous version of the logistic the  logistic map. map. The qualitative behavior is easily understood in terms of the phase the phase line: line: the derivative derivative is null when function function is unit 1

2

2

and the derivative is positive for f  for  f  between  between 0 and 1, and negative negative for f  for f abo above ve 1 or less less than than 0 (thou (though gh nega negati tive ve poppopulations do not generally accord with a physical model). This This yiel yields ds an uns unstab table le equil equilibr ibriu ium m at 0, and and a stabl stablee equiequilibrium at 1, and thus for any function value greater than zero and less than unit, it grows to unit. The above equation equation can be rewrit rewritten ten in the follo ollowing wing steps:

d f (x) =  f (x)(1  −  f (x)) dx dy =  y (1 −  y ) dx dy =  y  −  y 2 dx dy 2 − y  =  − y dx Which is a special case of the  Bernoulli differential equation and tion  and has the following solution:

f (x) =

APPLICA APPLICATIO TIONS  NS 

ex · 1 −  e −2x ex − e −x = tanh(x) = x e + e −x ex · (1 +  e −2x )

�

=  f (2x)  −

�

e−2x e−2x + 1  −  1 = (2 )  − = 2f (2x)  −  1.  f  x 1 +  e−2x 1 +  e−2x

The hyperbolic tangent relationship leads to another form for the logistic function’s derivative:

1 d f (x) =  sech2 4 dx

x

�� 2

,

which ties the logistic function into the  logistic distribution.. tion

1.3

Rotatio Rotational nal symmet symmetry ry about about (0, ½)

The sum of the logistic function and its reflection about the vertical axis, f  axis,  f  (−  (− x   x ) is

1 1 (1 +  ex ) + (1 + e−x ) 2 +  ex + e−x + = = 1 +  e−x 1 +  e−(−x) (1 +  e−x )(1 + ex ) 1 + ex + e−x + ex−

ex ex + C 

Choosing the constant of integration C  = 1  gives the other well-known form of the definition of the logistic curve

ex

1 = f (x) = x 1 + e−x e +1

The logisti logisticc fun functi ction on is thus rotation rotationall ally y symmetri symmetrical cal [4] about the point (0, 1/2).

2 2.1 2.1

Appl Appliicati cation onss In eco ecology: gy:

modeling ling populat lation

More More quant quantita itati tive vely ly,, as can can be seen seen from the analy analytic tical al sogrowth lution, the logistic curve shows early   exponential growth for negative argument, which slows to linear growth of A typic typical al appl applic icati ation on of the logi logisti sticc equat equatio ion n is a commo common n slope 1/4 for an argument near zero, t hen approaches approaches one model of population of  population growth (see growth (see also population also  population dynamwith an exponentially exponentially decaying decaying gap. ics), ics ), originally due to  to   Pierre-François Verhulst in Verhulst  in 1838, The logistic function is the inverse of the natural   logit where the rate of reproduction is proportional to both function and so can be used to convert the logarithm of the existing population and the amount of available reodds into odds  into a probability a  probability.. In mathematical notation the lo- sources, all else being equal. The Verhulst Verhulst equation was gistic gistic func unctio tion n is someti sometime mess writte written n as expit [3] in the the same same published after Verhulst had read  Thomas Malthus' Malthus' An form as logit  as logit . The conversion from the log-likelihood the  log-likelihood ra- Essay on the Principle of Population. Population. Verhulst derived his tio of tio  of two alternatives also takes the form of a logistic logistic equation to describe the self-limiting growth of curve. a  biological  biological   population. population. The equation was rediscovered rediscovered in 1911 by A. by  A. G. McKendrick for McKendrick  for the growth of bacteria The logistic sigmoid function is related to the hyperbolic the  hyperbolic in broth and experimentally tested using a technique for tangent,, A.p. by tangent nonlinear parameter estimation.[5] The equation is also sometimes called the  Verhulst-Pearl equation  following its rediscovery in 1920 by  Raymond Pearl (1879–1940) Pearl  (1879–1940) x 2 f (x) = 1 +  tanh and Lowell Low ell Reed (1888– (18 88–196 1966) 6) of the Johns John s Hopk Hopkins ins Uni Uni-2 [6] versity.. Another scientist, versity scientist, Alfred  Alfred J. Lotka derived Lotka  derived the or equation again in 1925, calling it the  law of population  growth.  growth.

��

tanh(x) = 2f (2x)  −  1 The latter relationship follows from

Letting P   P   represent population size (N  ( N    is often used in ecology instead) and t  and  t  represent  represent time, this model is formalized by the differential the  differential equation: equation:

2.2

3

In statistics and machine learning

lim P (t) =  K 

t→∞

Which is to say that K  is the limiting value of P : the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value  P (0) > 0, and also in the case that P (0) > K . In   ecology,   species   are sometimes referred to as r strategist or K -strategist  depending upon the   selective processes that have shaped their   life history   strategies. Choosing the variable dimensions so that n measures the population in units of carrying capacity, and τ  measures time in units of 1/r , gives the dimensionless differential equation dn =  n (1 −  n) dτ  2.1.1

Time-varying carrying capacity

Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying: K (t ) > 0, leading to the following mathematical model: Pierre-François Verhulst (1804–1849)

dP  =  rP  · dt

�

P  1  − K (t)

�

A particularly important case is that of carrying capacity that varies periodically with period T : dP  =  rP  · dt

�

P  1 − K 

�

where the constant r  defines the growth rate and K  is the carrying capacity. In the equation, the early, unimpeded growth rate is modeled by the first term +rP . The value of the rate r  represents the proportional increase of the population P  in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is −rP 2 /K ) becomes almost as large as the first, as some members of the population P  interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the  bottleneck , and is modeled by the value of the parameter K . The competition diminishes the combined growth rate, until the value of P  ceases to grow (this is called  maturity of the population). The solution to the equation (with P 0   being the initial population) is

P (t) = where

KP 0 ert K  +  P 0 (ert − 1)

K (t + T ) =  K (t) It can be shown that in such a case, independently from the initial value P (0) > 0, P (t ) will tend to a unique periodic solution P *(t ), whose period is T . A typical value of T  is one year: In such case K (t ) may reflect periodical variations of weather conditions. Another interesting generalization is to consider that the carrying capacity K (t ) is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation,[7] which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

2.2

In statistics and machine learning

Logistic functions are used in several roles in   statistics. For example, they are the  cumulative distribution func-

4

2

tion of the logistic family of distributions, and they are, a bit simplified, used to model the chance a chess player has to beat his opponent in the  Elo rating system. More specific examples now follow. 2.2.1

The logistic function is itself the derivative of another proposed activation function, the softplus.

2.3

In medicine: modeling of growth of tumors

Logistic regression

Main article: Logistic regression Logistic functions are used in logistic regression to model how the probability p of an event may be affected by one or more  explanatory variables: an example would be to have the model

APPLICATIONS 

See also: Gompertz curve § Growth of tumors Another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors. This application can be considered an extension of the above-mentioned use in the framework of ecology (see also the Generalized logistic curve, allowing for more parameters). Denoting with X (t ) the size of the tumor at time t , its dynamics are governed by:

 p  =  f (a + bx) where x  is the explanatory variable and a and b are model parameters to be fitted and f  is the standard logistic function. Logistic regression and other log-linear models are also commonly used in machine learning. A generalisation of the logistic function to multiple inputs is the softmax activation function, used in multinomial logistic regression. Another application of the logistic function is in the Rasch model, used in   item response theory. In particular, the Rasch model forms a basis for  maximum likelihood estimation of the locations of objects or persons on a continuum, based on collections of categorical data, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect. 2.2.2

Neural networks

Logistic functions are often used in  neural networks to introduce  nonlinearity in the model and/or to clamp signals to within a specified  range. A popular neural net element computes a linear combination of its input signals, and applies a bounded logistic function to the result; this model can be seen as a “smoothed” variant of the classical threshold neuron. A common choice for the activation or “squashing” functions, used to clip for large magnitudes to keep the response of the neural network bounded[8] is

g (h) =

1 1 + e−2βh

which is a logistic function. These relationships result in simplified implementations of artificial neural networks with   artificial neurons. Practitioners caution that sigmoidal functions which are  antisymmetric about the origin (e.g. the hyperbolic tangent) lead to faster convergence when training networks with backpropagation.[9]

′

X  =  r

�

X  1 − K 

�

X 

which is of the type:

X ′ =  F (X )X,

F ′ (X )  ≤  0

where F (X ) is the proliferation rate of the tumor. If a chemotherapy is started with a log-kill effect, the equation may be revised to be

′

X  =  r

�

X  1 − K 

�

X  − c(t)X,

where c (t ) is the therapy-induced death rate. In the idealized case of very long therapy, c (t ) can be modeled as a periodic function (of period T ) or (in case of continuous infusion therapy) as a constant function, and one has that

1 T 

T 

∫  0

c(t) dt > r → lim x(t) = 0 t→+∞

i.e. if the average therapy-induced death rate is greater than the baseline proliferation rate then there is the eradication of the disease. Of course, this is an oversimplified model of both the growth and the therapy (e.g. it does not take into account the phenomenon of clonal resistance).

2.4

In chemistry: reaction models

The concentration of reactants and products in autocatalytic reactions follow the logistic function.

2.5

In physics: Fermi distribution

The logistic function determines the statistical distribution of fermions over the energy states of a system in thermal equilibrium. In particular, it is the distribution of the

5

probabilities that each possible energy level is occupied by a fermion, according to Fermi–Dirac statistics.

 frenzy, the rapid build out as synergy and the completion as maturity.[15]

2.6

3

In linguistics: language change

In linguistics, the logistic function can be used to model language change:[10] an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted.

See also •   Diffusion of innovations • Generalised logistic curve •  Gompertz curve

2.7

In economics and sociology: diffusion of innovations

•   Heaviside step function •  Hubbert curve

The logistic function can be used to illustrate the progress of the diffusion of an innovation through its life cycle.

•  Logistic distribution •  Logistic map

In The Laws of Imitation (1890), Gabriel Tarde describes the rise and spread of new ideas through imitative chains. In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with f (x) = 2x ; finally, the third stage is logarithmic, with f (x) =  log (x) , and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear. The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote. In the history of economy, when new products are introduced there is an intense amount of research and development which leads to dramatic improvements in quality and reductions in cost. This leads to a period of rapid industry growth. Some of the more famous examples are: railroads, incandescent light bulbs,   electrification, cars and air travel. Eventually, dramatic improvement and cost reduction opportunities are exhausted, the product or process are in widespread use with few remaining potential new customers, and markets become saturated. Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis (IIASA). These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle. Long economic cycles were investigated by Robert Ayres (1989). [11] Cesare Marchetti published on   long economic cycles and on diffusion of innovations.[12][13] Arnulf Grübler’s book (1990) gives a detailed account of the diffusion of infrastructures including canals, railroads, highways and airlines, showing that their diffusion followed logistic shaped curves.[14] Carlota Perez used a logistic curve to illustrate the long (Kondratiev) business cycle with the following labels: beginning of a technological era as  irruption, the ascent as

•   Logistic regression • Logistic smooth-transmission model •   Logit •  Log-likelihood ratio •  Malthusian growth model •   Population dynamics •  r/K selection theory •  Shifted Gompertz distribution •  Tipping point (sociology) • Rectifier (neural networks)

4

Notes

[1] Verhulst, Pierre-François (1838). “Notice sur la loi que la population poursuit dans son accroissement” (PDF). Correspondance mathématique et physique. 10: 113–121. Retrieved 3 December 2014. [2] Verhulst, Pierre-François (1845).  “Recherches mathématiques sur la loi d'accroissement de la population” [Mathematical Researches into the Law of Population Growth Increase]. Nouveaux Mémoires de l'Académie Royale des  Sciences et Belles-Lettres de Bruxelles . 18: 1–42. Retrieved 2013-02-18. [3] expit documentation for R’s clusterPower package [4] Raul Rojas.  Neural Networks - A Systematic Introduction (PDF). Retrieved 15 October 2016. [5] A. G. McKendricka; M. Kesava Paia1 (January 1912). “XLV.—The Rate of Multiplication of Microorganisms: A Mathematical Study”.   Proceedings  of the Royal Society of Edinburgh. 31: 649–653. doi:10.1017/S0370164600025426.

6

6

[6]   Raymond Pearl and  Lowell Reed   (June 1920).   “On the Rate of Growth of the Population of the United States” (PDF). Proc. of the National Academy of Sciences . 6 (6). p. 275. [7] Yukalov, V. I.; Yukalova, E. P.; Sornette, D. (2009). “Punctuated evolution due to delayed carrying capacity”.  Physica D: Nonlinear Phenomena. 238 (17): 1752. doi:10.1016/j.physd.2009.05.011.

[9] LeCun, Y.; Bottou, L.; Orr, G.; Muller, K. (1998). Orr, G.; Muller, K., eds.  Efficient BackProp (PDF). Neural Networks: Tricks of the trade. Springer. ISBN 3-540-653112. [10] Bod, Hay, Jennedy (eds.) 2003, pp. 147–156 [11]   Ayres, Robert   (1989).   “Technological Transformations and Long Waves” (PDF). [12] Marchetti, Cesare (1996). “Pervasive Long Waves: Is Society Cyclotymic” (PDF). [13] Marchetti, Cesare (1988).   “Kondratiev Revisited-After One Cycle” (PDF). [14] Grübler, Arnulf (1990). The Rise and Fall of Infrastructures: Dynamics of Evolution and Technological Change in Transport   (PDF). Heidelberg and New York: PhysicaVerlag. [15] Perez, Carlota (2002). Technological Revolutions and Financial Capital: The Dynamics of Bubbles and Golden Ages . UK: Edward Elgar Publishing Limited.   ISBN 184376-331-1.

References • Jannedy, Stefanie;

Bod, Rens; Hay, Jennifer (2003).  Probabilistic Linguistics . Cambridge, Massachusetts: MIT Press.  ISBN 0-262-52338-8.

•  Gershenfeld, Neil A. (1999).  The Nature of Math-

ematical Modeling. Cambridge, UK: Cambridge University Press.  ISBN 978-0-521-57095-4. • Kingsland, Sharon E. (1995).

Modeling nature: episodes in the history of population ecology. Chicago: University of Chicago Press.  ISBN 0-22643728-0.

• Weisstein,

Eric

W.

“Logistic

Equation”.

MathWorld .

6

External links •   L.J. Linacre,  Why logistic ogive and not autocat-

alytic curve?, accessed 2009-09-12. •  http://luna.cas.usf.edu/~{}mbrannic/files/

regression/Logistic.html

Eric

W.

“Sigmoid

Function”.

MathWorld . •  Online experiments with JSXGraph •  Esses are everywhere. •  Seeing the s-curve is everything. • Restricted Logarithmic Growth with Injection

[8] Gershenfeld 1999, p.150

5

• Weisstein,

EXTERNAL LINKS 

7

7

Text and image sources, contributors, and licenses

7.1 •

7.2 •

•

7.3 •

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Hyperbolic function “Sech” redirects here. For other uses, see Sech (disambiguation). “Sinh” redirects here. For the garment, see sinh (clothing). “Hyperbolic curve” redirects here. For the geometric curve, see Hyperbola.

Hyperbolic functions occur in the solutions of many linear differential equations, for example the equation defining a catenary, of some cubic equations, in calculations of angles and distances in  hyperbolic geometry  and of Laplace’s equation in Cartesian coordinates.   Laplace’s equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence holomorphic.

Y  cosh a

 x² – y² = 1

sinh 1 1

a

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[9] Riccati used Sc.  and  Cc.  ( [co]sinus circulare ) to refer to circular functions and  Sh. and  Ch. ([co]sinus  hyperbolico ) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.[10] The abbreviations sh  and ch  are still used in some other languages, like French and Russian.

a/2  X 

A ray through the  unit hyperbola x2 − y2 = 1   in the point  (cosh a, sinh a)  , where a  is twice the area between the ray, the hy perbola, and the  x  -axis. For points on the hyperbola below the  x -axis, the area is considered negative (see  animated version with comparison with the trigonometric (circular) functions).

1

Standard analytic expressions

In mathematics, hyperbolic functions  are analogs of the ordinary trigonometric, or circular functions. The basic hyperbolic functions are the  hyperbolic sine “sinh” (/ˈsɪntʃ/ or  /ˈʃaɪn/),[1] and the  hyperbolic cosine “cosh” (/ˈkɒʃ/),[2] from which are derived the  hyperbolic tangent “tanh” (/ˈtæntʃ/ or /ˈθæn/),[3] hyperbolic cosecant  “csch” or “cosech” (/ˈkoʊʃɛk/[2] or /ˈkoʊsɛtʃ/), hyperbolic secant  “sech” (/ˈʃɛk/ or  /ˈsɛtʃ/),[4] and   hyperbolic cotangent  “coth” (/ˈkoʊθ/ or /ˈkɒθ/),[5][6] corresponding to the derived trigonometric functions. The inverse hyperbolic functions  are the   area hyperbolic sine  “arsinh” (also called “asinh” or sometimes “arcsinh”) [7][8] and so on. Just as the points (cos  t , sin  t ) form a circle with a unit radius, the points (cosh  t , sinh t ) form the right half of the equilateral hyperbola. The hyperbolic functions take sinh, cosh and tanh a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions are: The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. •  Hyperbolic sine: 1

2

1

STANDARD ANALYTIC EXPRESSIONS 

csch, sech and coth

(b) sinh(x) is half the difference of  e x and  e−x

•  Hyperbolic tangent:   sinh x ex − e = tanh x  = cosh x ex + e

−x −x

=

=

1 e−2x e2x 1 = 1 + e−2x e2x + 1

−

−

• Hyperbolic cotangent: x ̸= 0  cosh x ex + e coth x  = = sinh x ex − e

−x

x

(a) cosh(x) is the average of  e and  e

−x

−x

=

sinh x  =

ex

−x

−e 2

=

1 e−2x e2x 1 = 2ex 2e−x

−

−

1 + e−2x e2x + 1 = 1 e−2x e2x 1

−

sech x  = cosh x  =

2

1 + e−2x e2x + 1 = = 2ex 2e−x

−

•  Hyperbolic secant:

•  Hyperbolic cosine: ex + e−x

=

1

cosh x

=

2 = ex + e−x

2ex 2e−x = 2x = 1 + e−2x e +1

3

2

• Hyperbolic cosecant:  x ̸= 0

Special meanings

2.1

csch  x  = =

1

sinh x

2ex e2x

−1

=

2

=

x

e

−x

−e

=

2e−x 1

−e

Hyperbolic cosine

It can be shown that the area under the curve of cosh ( x ) over a finite interval is always equal to the arc length corresponding to that interval:[11]

2x

−

b

∫ 

Hyperbolic functions can be introduced via  imaginary area  = circular angles:

•  Hyperbolic sine:

2.2

∫  �  � b

cosh (x) dx =

a

1+

a

d  cosh (x) dx

2

�

dx  =  length arc

Hyperbolic tangent

The hyperbolic tangent is the solution to the  differential equation f  = 1 − f 2 with f(0)=0 and the  nonlinear boundary value problem:[12] [13] ′

sinh x  = −i sin(ix)

•  Hyperbolic cosine: cosh x  =  cos(ix)

•  Hyperbolic tangent: tanh x  = −i tan(ix)

1 ′′ f  =  f 3 2

3

f (0) =  f ′ (

− f ;

∞) = 0

Useful relations

Odd and even functions: sinh(−x) = − sinh x cosh(−x) = cosh x Hence:

• Hyperbolic cotangent: coth x  =  i cot(ix)

• Hyperbolic secant: sech x  =  sec(ix)

• Hyperbolic cosecant:

tanh(−x) = − tanh x coth(−x) = − coth x sech(−x) = sech x csch(−x) = − csch x It can be seen that cosh  x  and sech  x  are even functions; the others are odd functions.

arsech x  =  arcosh arcsch x  =  arsinh

where  i  is the imaginary unit with the property that  i 2 = −1. The complex forms in the definitions above derive from Euler’s formula.

x

 1

arcoth x  =  artanh csch x  =  i csc(ix)

 1

x

 1 x

Hyperbolic sine and cosine satisfy: cosh x + sinh x  =  e x cosh x − sinh x  =  e x cosh2 x − sinh2 x  = 1 −

4

5

the last which is similar to the Pythagorean trigonometric identity.

tanh [15]

One also has

4

sech2 x  = 1 − tanh2 x csch2 x  =  coth2 x − 1

DERIVATIVES 

(x)−1 =   cosh sinh(x) =   coth(x)

x

 2

− csch(x)

Inverse functions as logarithms

Main article: Inverse hyperbolic function

for the other functions. 3.1

Sums of arguments

� √  � � √  − � ≥ � �|| � − �|| � −�  � �  − � �  � �

arsinh(x) = ln x +

x2 + 1

arcosh(x) = ln x +

x2

1 2 1 arcoth(x) =  ln 2

sinh(x + y ) = sinh(x) cosh(y) + cosh(x) sinh(y ) cosh(x + y ) = cosh(x) cosh(y ) + sinh(x) sinh(y )   tanh x + tanh y tanh(x + y ) = 1 + tanh x tanh y particularly

1+x 1 x x+1 x 1

artanh(x) =  ln

arsech(x) = ln

1

x

+

cosh(2x) = sinh2 x + cosh2 x  = 2 sinh2 x + 1 = 2 cosh2 x − 1 1 + arcsch(x) = ln sinh(2x) = 2 sinh x cosh x x

1 ;x

1

; x <  1 ; x >  1

1

1

x2

1

x2

+1

= ln = ln

√  1 + 1 − x2 x

1+

√ x2 + 1 x

Also:  x + y

5

−y sinh x + sinh y  = 2 sinh   cosh 2 2  x + y  x − y cosh x + cosh y  = 2 cosh   cosh  x

2

3.2

d  sinh x  =  cosh x dx

2

d  cosh x  =  sinh x dx

Subtraction formulas

sinh(x − y ) = sinh(x) cosh(y) − cosh(x) sinh(y ) cosh(x − y ) = cosh(x) cosh(y ) − sinh(x) sinh(y ) Also:  x + y

−y 2 2  x + y  x − y cosh x − cosh y  = 2 sinh   sinh sinh x − sinh y  = 2 cosh

  sinh

 x

2

2

Source.[14] 3.3

Half argument formulas

  sinh(x) = sgn(x) 2 2(cosh(x) + 1) where  sgn is the sign function. sinh

x

� � √ 

cosh

=

� � �  �� x

2

=

� 

cosh(x) −

d  tanh x  = 1 dx

− tanh2 x =  sech2 x = 1/ cosh2 x

d  coth x  = 1 dx

− coth2 x = − csch2 x = −1/ sinh2 x

d   sech x  = dx

− tanh x   sech  x

d   csch x  = dx

− coth x  csch x

d  arsinh  x  = dx

√ x21+ 1

d  arcosh x  = 1 dx

√ x21− 1

2

1 d   artanh x  = 1 x2 dx

−

1 d   arcoth  x  = 1 x2 dx

cosh(x) + 1 2

  sinh(x) = = sgn(x) tanh cosh(x) + 1 2 If  x  ≠ 0, then x

Derivatives

� 

cosh(x) − 1 ex = cosh(x) + 1 ex

− 1 d  arsech  x  = − √  x 1 − x2 −dx1 +d 1 1  arcsch  x  = − √  dx |x| 1 + x2

� �

; 0  < x ;x = 0

̸

≤1

5

6

Second derivatives x3

x5

∞

x7

x2n+1

∑

+ + + · ·· = Sinh and cosh are both equal to their second derivative, sinh x  =  x + 3! 5! 7! (2n + 1)! n=0 that is:

The function sinh x   has a Taylor series expression with only odd exponents for  x . Thus it is an odd function, that is, −sinh  x  = sinh(− x ), and sinh 0 = 0.

d2  sinh x  =  sinh x dx2 d2  cosh x  =  cosh x . dx2

cosh x  = 1 +

 x 2

2!

+

x4

4!

x6

+

6!

∞

+

·· · =

∑

n=0

x2n (2n)!

All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions The function cosh  x  has a Taylor series expression with only even exponents for  x . Thus it is an even function, ex and e x , and the zero function  f (x) = 0 . that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of the exponential function. −

7

Standard integrals

For a full list, see list of integrals of hyperbolic functions.

∫  ∫  ∫  ∫  ∫  ∫ 

  sinh(ax) dx =  a

  cosh(ax) dx =  a   tanh(ax) dx =  a

  coth(ax) dx =  a

1

−

1

−

1

−

1

−

cosh(ax) + C 

coth x  =  x

ln(cosh(ax)) + C  ln(sinh(ax)) + C 

ln tanh

−

ax

� � ��

The proved

2

following integrals using   hyperbolic 1 √  2 2  du =  arsinh u + C  a +u

1

u2

a2

1

a2

u2

1

a2

u2

1

u a2

u2

1

u

a2

+

u2

a u a

 du  =  a −1 arcoth

 du  =

1

−

1

−

−a

arsech arcsch

u a u a

+ C ; u2 < a2 + C ; u2 > a2

where  C  is the constant of integration.

8

Taylor series expressions

+

−

x3

x

17x7 + 315

∞

·· · =

 2 x5

5x4 + 2 24

−

 7 x3

x

61x6 + 720 31x5

1

−

|

−

E n  is

9

n

x2n−1

(2n)!

,

∞

22n B2n x2n−1 , 0  < (2 n)! n=1

∑

+

∞

· ·· =

− 1)B2

n=1

 − 45 + 945 + · · · =  x 3

x2

22n (22n

∑

∑

n=0

− 6 + 360 − 15120 + · ·· = x

E 2n x2n π , x < (2n)! 2

||

∞

1

−

+

∑

n=1

=  a −1 ln csch(B coth ax) is the ax) + C  number  n(th Bernoulli n

can be substitution:

u + C  a u + C  a

1

−

2x5 + 3 15

where:

+ C 

 du  =  a −1 artanh

−a

+ C 

�� �� �� �� ��  

 du  =  arcosh

 du  =

csch  x  =  x

1

  csch(ax) dx =  a

∫  ∫  √  ∫  − ∫  − ∫  √  − ∫  √  −

sech  x  = 1 −

arctan(sinh(ax)) + C 

−

1

−

sinh(ax) + C 

1

  sech(ax) dx =  a

tanh x  =  x −

x3

|

the  nth Euler number

Comparison with circular functions

The hyperbolic functions represent an expansion of trigonometry   beyond the   circular functions. Both types depend on an   argument, either circular angle or hyperbolic angle. Since the area of a circular sector with radius r  and angle u  is r 2u ,  it will be equal to  u  when  r  = square root of 2. In the diagram such a circle is tangent to the hyperbola  xy = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red sectors together depict an area and hyperbolic angle magnitude. 2

The legs of the two right triangles with hypotenuse on the ray defining the angles are of length √2 times the circular and hyperbolic functions.

It is possible to express the above functions as Taylor se- Mellon Haskell of University of California, Berkeley deries: scribed the basis of hyperbolic functions in areas of

2(1

− 22

n−1

)B (2n)!

6

11

 y

 y=1/x  y=x

 x 2+y 2=2

RELATIONSHIP TO THE EXPONENTIAL FUNCTION 

   )    u    h  (    s   c  o̅                 2    )    √    u   (    s   c  o                 2 ̅    √

u

√      

2     ̅        s  i    n  h   √       (    2    u     ̅         )   s  i    n  (    u     )  

 y=ax: a<1

 x

sinh(2x) = 2 sinh x cosh x cosh(2x) = cosh2 x + sinh2 x  = 2 cosh2 x − 1 = 2 sinh2 x + 1 2 tanh x tanh(2x) = 1 + tanh2 x 2 tanh x sinh(2x) = 1 − tanh2 x 1 + tanh2 x cosh(2x) = 1 − tanh2 x and the “half-argument formulas”[17]

� 

1 ( 1)   Note: This is sinh 2 = 2 cosh x − equivalent to its circular counterpart multiplied by −1. x

�  � 

cosh 2 = 12 (cosh x + 1)  Note: This corresponds to its circular counterpart. x

Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of  circular sector  area  u  and hyperbolic   functions depending on  hyperbolic sector   area u.

x 1 tanh x2 = cosh cosh x+1 = coth x − csch x. −

  sinh x cosh x+1

x−1 =   cosh sinh x =

coth x2 = coth x + csch x. hyperbolic sectors in an 1895 article in Bulletin of the American Mathematical Society (see External links). He refersto the hyperbolic angle as an invariant measurewith The derivative of sinh x   is cosh  x  and the derivative of respect to the squeeze mapping just as circular angle is cosh x  is sinh x ; this is similar to trigonometric functions, invariant under rotation. albeit the sign is different (i.e., the derivative of cos  x  is −sin  x ). The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.

10

Identities

The graph of the function  a cosh( x /a) is the catenary, the curve formed by a uniform flexible chain hanging freely between two fixed points under uniform gravity.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact,   Osborn’s rule [16] states that one can convert any 11 Relationship to the exponential trigonometric identity into a hyperbolic identity by exfunction panding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a prod- From the definitions of the hyperbolic sine and cosine, we uct of 2, 6, 10, 14, ... sinhs. This yields for example the can derive the following identities: addition theorems ex = cosh x + sinh x

sinh(x + y ) = sinh(x) cosh(y) + cosh(x) sinh(y ) cosh(x + y ) = cosh(x) cosh(y ) + sinh(x) sinh(y )   tanh(x) + tanh(y ) tanh(x + y ) = 1 + tanh(x) tanh(y ) the “double argument formulas”

and

e−x = cosh x

− sinh x

These expressions are analogous to the expressions for sine and cosine, based on  Euler’s formula, as sums of complex exponentials.

7

12

Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z  and cosh  z  are then holomorphic.

[3]  Collins Concise Dictionary , p. 1520 [4]  Collins Concise Dictionary , p. 1340 [5]  Collins Concise Dictionary , p. 329 [6]   tanh [7]   Abramowitz, Milton;   Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York: Dover Pub-

Relationships to ordinary trigonometric functions are given by Euler’s formula for complex numbers:

lications, ISBN 978-0-486-61272-0 [8] Some examples of using arcsinh found in Google Books.

eix = cos x + i

sin x e ix = cos x − i sin x

[9] Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer.  Euler at 300: an appreciation.  Mathematical Association of America, 2007. Page 100.

−

so:

[10] Georg F. Becker.   Hyperbolic functions.   Read Books, 1931. Page xlviii.

1 ix e + e−ix = cos x 2 1 sinh(ix) = eix e−ix =  i sin x 2 cosh(x + iy ) = cosh(x) cos(y) + i sinh(x) sin(y ) sinh(x + iy ) = sinh(x) cos(y ) + i cosh(x) sin(y )

cosh(ix) =

 

−

 

tanh(ix) = i tan x cosh x  =  cos(ix) sinh x  = −i sin(ix) tanh x  = −i tan(ix)

[11] N.P., Bali (2005).  Golden Integral Calculus . Firewall Media. p. 472. ISBN 81-7008-169-6. [12]  Eric W. Weisstein.  “Hyperbolic Tangent”.   MathWorld. Retrieved 2008-10-20. [13]  “Derivation of tanh solution to 1/2 f′′=f^3−f...”.   Math StackExchange. Retrieved 18 March 2016. [14] Martin, George E. (1986).  The foundations of geometry and the non-euclidean plane  (1., corr. Springer ed.). New York: Springer-Verlag. p. 416. ISBN 3-540-90694-0. [15]   "math.stackexchange.com/q/1565753/88985". StackExchange   (mathematics). Retrieved 24 January 2016.

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period  2 πi  ( πi  for hyperbolic tangent and cotangent). [16] Osborn, G. (July 1902). “Mnemonic for hyperbolic formulae”.  The Mathematical Gazette. 2 (34): 189. JSTOR 3602492.

13

See also

• e (mathematical constant) •  Equal incircles theorem, based on sinh •  Inverse hyperbolic functions •  List of integrals of hyperbolic functions •  Poinsot’s spirals •  Sigmoid function •  Trigonometric functions •  Modified hyperbolic tangent 14

References

[1] (1999)   Collins Concise Dictionary , 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386 [2]  Collins Concise Dictionary , p. 328

[17] Peterson, John Charles (2003). Technical mathematics  with calculus  (3rd ed.). Cengage Learning. p. 1155. ISBN 0-7668-6189-9., Chapter 26, page 1155

15

External links

•  Mellen W. Haskell (1895) On theintroduction of the notion of hyperbolic functions Bulletin of the American Mathematical Society 1(6):155–9.

• Hazewinkel,

Michiel, ed.

(2001),   “Hyperbolic functions”,  Encyclopedia of Mathematics , Springer, ISBN 978-1-55608-010-4

•  Hyperbolic functions on PlanetMath •  Hyperbolic functions entry at MathWorld •  GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)

•  Web-based calculator of hyperbolic functions

8

16

16

Text and image sources, contributors, and licenses

16.1 •

TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

Text

  Hyperbolic function Source:  https://en.wikipedia.org/wiki/Hyperbolic_function?oldid=776492647 Contributors:  AxelBoldt, The Anome,

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16.2 •

•

•

•

•

•

•

•

•

•

•

•

•

•

File:Circular_and_hyperbolic_angle.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/c/c6/Circular_and_hyperbolic_ angle.svg License:  CC0  Contributors:  Own work  Original artist:  Original: Maschen   File:Commons-logo.svg  Source:   https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License:  PD  Contributors:  ?  Original artist:  ?   File:Complex_Cosh.jpg   Source:    https://upload.wikimedia.org/wikipedia/commons/4/4e/Complex_Cosh.jpg   License:    Public Contributors:  Eigenes Werk (own work) made with mathematica 5.0  Original artist:  Jan Homann

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16.3 •

Images

Content license

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Inverse trigonometric functions In  mathematics, the   inverse trigonometric functions (occasionally called   cyclometric functions[1] ) are the inverse functions  of the  trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

2.1

Principal values

Since none of the six trigonometric functions are one-toone, they are restricted in order to have inverse functions. Therefore the ranges of the inverse functions are proper subsets of the domains of the original functions

For example, using  function  in the sense of multivalued functions, just as the square root function y = √ x  could be defined from  y2 =  x , the function  y  = arcsin( x ) is defined so that sin(y) = x . There are multiple numbers y  such that sin(y) =  x ; for example, sin(0) = 0, but also sin(π) = 1 Notation 0, sin(2π) = 0, etc. When only one value is desired, the function may be restricted to its principal branch. With There are several notations used for the inverse trigonothis restriction, for each  x  in the domain the expression metric functions. arcsin( x ) will evaluate only to a single value, called its The most common convention is to name inverse trigono- principal value. These properties apply to all the inverse metric functions using an arc- prefix, e.g., arcsin( x ), ar- trigonometric functions. ccos( x ), arctan( x ), etc. This convention is used throughThe principal inverses are listed in the following table. out the article. When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, (Note: Some authors define the range of arcsecant to be ( where r is the radius of the circle. Thus, in the unit cir- 0 ≤ y <π/2orπ≤ y < 3π/2 ), because thetangent function cle, “the arc whose cosine is x” is the same as “the an- is nonnegative on this domain. This makes some comgle whose cosine is x”, because the length of the arc of putations more consistent. For example using this range, the circle in radii is the same as the measurement of the tan(arcsec( x )) = √ x 2 − 1, whereas with the range ( 0 ≤ y < angle in radians.[2] Similarly, in computer programming π/2orπ/2< y ≤ π ), wewould haveto write tan(arcsec( x )) languages the inverse trigonometric functions are usually = ±√ x 2 − 1, since tangent is nonnegative on 0 ≤  y  < π/2 but nonpositive on π/2 <  y  ≤ π. For a similar reason, the called asin, acos, atan. same authors define the range of arccosecant to be −π < The notations sin−1 ( x ), cos−1 ( x ), tan−1 ( x ), etc., as introy ≤ −π/2 or 0 <  y  ≤ π/2.) duced by John Herschel in 1813,[3][4] are often used as well, but this convention logically conflicts with the com- If x  is allowed to be a complex number, then the range of mon semantics for expressions like sin 2 ( x ), which refer y applies only to its real part. to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and  compositional inverse. The confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, (cos( x ))−1 = sec( x ). Nevertheless, certain authors 2.2 Relationships between trigonometric advise against using it for its ambiguity.[5] functions and inverse trigonometric Another convention used by a few authors [6] is to use functions a majuscule (capital/upper-case) first letter along with a −1 superscript, e.g., Sin−1 ( x ), Cos−1 ( x ), Tan−1 ( x ), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin −1 ( x ), cos−1 ( x ), Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them etc. is by considering the geometry of a right-angled triangle, with one side of length 1, and another side of length  x   (any real number between 0 and 1), then applying the 2 Basic properties Pythagorean theorem and definitions of the trigonometric ratios. Purely algebraic derivations are longer. 1

2

2

BASIC PROPERTIES 

Principal values of the arcsec(x) andarccsc(x) functions graphed  on the cartesian plane.

arccos(x) = arccot(x) = arccsc(x) =

π

2 π

2 π

2

 − arcsin(x)  − arctan(x)  − arcsec(x)

Negative arguments:

The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.

arcsin(−x) = − arcsin(x) arccos(−x) = π − arccos(x) arctan(−x) = − arctan(x) arccot(−x) = π − arccot(x) arcsec(−x) = π − arcsec(x) arccsc(−x) = − arccsc(x) Reciprocal arguments:

arccos arcsin arctan arctan arccot The usual principal values of the arctan( x) and arccot(x) functions graphed on the cartesian plane.

arccot arcsec

2.3

Relationships among the trigonometric functions

Complementary angles:

inverse

arccsc

�� �� �� �� �� �� �� �� 1

x

1

x

1

x

1

x

1

x

1

x

1

x

1

x

= arcsec(x) = arccsc(x) = = = =

π

2

 − arctan(x) = arccot(x) , if x > 0

− π2 − arctan(x) = arccot(x) − π , if x < 0 π

2

 − arccot(x) = arctan(x) , if x > 0

3π 2

− arccot(x) = π  + arctan(x) , if x < 0

= arccos(x) = arcsin(x)

If you only have a fragment of a sine table:

3.2

3

Expression as definite integrals 

arccos(x) = arcsin

�√  − �  − −  � √  � 2

1

x

1 2 1 arcsin(x) =  arccos 1 2

arccos(x) =  arccos 2x2

arctan(x) = arcsin

 ,

if 0 ≤ x ≤ 1

1  , if 0

2x2

≤x≤1  , if 0 ≤ x ≤ 1

x

2

x +1

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real). From the half-angle formula, tan

arcsin(x) = 2 arctan arccos(x) = 2 arctan arctan(x) = 2 arctan

2.4

θ

 2

=

sin θ  , 1+cos θ

we get:

√ 1 1− z

= =

2

− √ 1 1− z

;

z=

̸  −1, +1

2

;

z=

̸  −1, +1

1 ; 1 + z 2 1 = ; 1 +  z 2 1 = ; 1 2 z 1 z =

z=

̸  −i, +i z̸ = −i, +i z̸ = −1, 0, +1

−

�  − − �  −

2

d  arccsc(z ) = dz

1

z2

1

1

; z=

z2

̸  −1, 0, +1

Only for real values of  x :

� √  � − � √  − �  − � √  � x

1+

x2

1

1 x2 1 +  x

,

if

1  < x

≤ +1

1+

d 1 ;  arcsec(x) = dx x x2 1 d 1 ;  arccsc(x) = dx x x2 1

| |√  − − | |√  −

|x| > 1 |x| > 1

For a sample derivation: if θ  =  arcsin x , we get:

x

1 + x2

d arcsin(x) dθ = = dx d sin(θ)

Arctangent addition formula

arctan(u)+arctan(v) = arctan

�

u + v 1 uv

−

�

(mod  π ) ,

This is derived from the tangent addition formula

tan(α + β ) =

d  arcsin(z ) dz d  arccos(z ) dz d  arctan(z ) dz d  arccot(z ) dz d  arcsec(z ) dz

tan(α) + tan(β )  , 1 − tan(α) tan(β )

by letting

3.2

dθ = cos(θ)dθ

1 = cos(θ)

1

�  − 1

sin2 (θ)

̸ 

Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral: x

arcsin(x) =

1

�  √  �  √  − �  − �  �  √  �  √  −

z2

1

0

1

arccos(x) =

β  =  arctan(v ) .

1

dz ,

|x| ≤ 1

dz ,

|x| ≤ 1

1 z2 1  dz , arctan(x) = 2 0 z +1 ∞ 1  dz , arccot(x) = z2 + 1 x x

3 3.1

In calculus

arcsec(x) =

1

arccsc(x) =

1

z z2

∞

1

1

dz  =  π  +

−1

�  x

x

dz  =

� 

1 √  dz , z z −1 2

1 √  dz , z z −1

2 Derivatives of inverse trigonometric z z2 − 1 −∞ x functions When  x  equals 1, the integrals with limited domains are improper integrals, but still well-defined. Main article:   Differentiation of trigonometric functions 3.3 Infinite series

The derivatives for complex values of  z  are as follows:

√ 1 1

uv = 1 .

x x

α  =  arctan(u) ,

=

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as

x

≥1

x

≥1

4

3

follows. For arcsine, the series can be derived by expanding its derivative, √ 11−z  , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1+1z in a geometric series and applying the integral definition above (see  Leibniz series).

numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)2 , with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

2

2

3.4 z3

z5

z7

Indefinite integrals of inverse trigono∞ ∞ 2n z 2n+1 metric (2n −functions 1)!! z 2n+1

� � � · � � · · � ·· · � · · · �− −  − ·· · � | | ≤ �  − �  �    ∑ �  �  �∏ �  �  � �  � 

1 arcsin(z ) = z + 2

arctan(z ) = z

z3

3

1 3 + 3 2 4

+

z5

z7

5

7

1 3 5 + 5 2 4 6

+

=

7

Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos x = π /2 arcsin x , arccsc x =  arcsin(1/x) , and so on. Another series is given by: 2 arcsin  x2

2

x2 n=1 n2 (2

∞

=

n

)

[7]

n

n

+

=

Leonhard Euler found a more efficient series for the arctangent, which is:

(2n)!!

· 2n + 1 =

For real and complex values of  z : n=0

∞

( 1)n z 2n+1 ; 2 n + 1 n=0

IN CALCULUS 

z

1

z =  i,

̸ 

−i

� 

n n 4 (2n +

n=0

√  − √  − − −   �� − ��

  arcsin(z ) dz  =  z  arcsin(z ) +

1

z 2 + C 

  arccos(z ) dz  =  z  arccos(z )

1

z 2 + C 

∞

  arcsec(z ) dz  =  z  arcsec(z )

n=0 k

n

2kz 2  . (2 k + 1)(1 + z 2 ) =1

(Notice that the term in the sum for  n  = 0 is the empty product which is 1.) Alternatively, this can be expressed: ∞

22n (n!)2 z 2n+1 arctan z  = (2n + 1)! (1 +  z 2 )n+1 n=0 3.3.1

Variant: Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:

 ;

ln z 1 +

  �  − �� �  − �� z2 1 z2

+ C 

z2 1 z2

+ C 

For real  x  ≥ 1:

� √  − � � √  − �

  arcsec(x) dx  =  x  arcsec(x) − ln x +

x2

1 + C 

  arccsc(x) dx  =  x  arccsc(x) + ln x +

x2

1 + C 

For all real  x  not between −1 and 1:

� √  − � � √  − �

  arcsec(x) dx  =  x  arcsec(x) − sgn(x) ln x +

x2

1

+ C 

  arccsc(x) dx  =  x  arccsc(x) + sgn(x) ln x +

x2

1

+ C 

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosez z function is also necessary due cant functions. The signum = arctan(z ) = (1z )2 z )2 to the absolute values in(1the derivatives of the two func1+ 1 + 2 2 tions, which create two different (3z ) (2z ) solutions for positive and 3 − 1z 2 + 3 + negative values of x. These can be further simplified us(5z )2 (3z )2 2 ing the logarithmic definitions of the inverse hyperbolic 5 − 3z + 5+ 2 2 (7 z ) (4 z ) functions: 7 − 5z 2 + 7+  .  . . 9 − 7z 2 + . 9 + .. The second of these is valid in the cut complex plane. There are two cuts, from − i  to the point at infinity, going down the imaginary axis, and from  i  to the point at infinity, going up the same axis. It works best for real

|z | ≤ 1

 1  ln 1 +  z 2 + C  2 1   arccot(z ) dz  =  z  arccot(z ) +  ln 1 + z 2 + C  2

  arctan(z ) dz  =  z  arctan(z )

  arccsc(z ) dz  =  z  arccsc(z ) + ln z 1 +

z arctan(z ) = 1 +  z 2

1)

�  � 

  arcsec(x) dx  =  x  arcsec(x) − arcosh(|x|) + C    arccsc(x) dx  =  x  arccsc(x) + arcosh(|x|) + C 

4.1

5

Logarithmic forms 

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to π arccot(z ) =  − arctan(z ) z̸ = −i, i the signum logarithmic function shown above. 2 All of these antiderivatives can be derived using which has the same cut as arctan; integration by parts  and the simple derivative forms shown above. arcsec(z ) = arccos 3.4.1

Example

Using u dv  =  uv

∫ 

u  =  arcsin(x) du  = √  1−x

v du , set

arccsc(z ) = arcsin

v  =  x

2

1

z=

̸  −1, 0, +1

z

where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;

− ∫ 

dv  =  dx

dx

�� �� 1

z=

̸  −1, 0, +1

z

which has the same cut as  arcsec.

Then

� 

  arcsin(x) dx  =  x arcsin(x)

�  − √ 

4.1

x

dx

Logarithmic forms

which by a simple substitution yields the final result:

These functions may also be expressed using  complex logarithms. This extends in a natural fashion their domain to the complex plane.

� 

arcsin(z ) = −i ln iz +

1

arccos(z ) = −i ln z +

z2

  arcsin(x) dx  =  x arcsin(x) +

4

1

√  − 1

2

−x

x2 + C 

Extension to complex plane

Since the inverse trigonometric functions are   analytic functions, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points. One possible way of defining the extensions is: z

arctan(z ) =

�  0

dx 1 +  x2

z=

̸  −i, +i

where the part of the imaginary axis which does not lie strictly between −i and +i is the cut between the principal sheet and other sheets;

arcsin(z ) = arctan

� √  � z

1

−z

2

z=

̸  −1, +1

� √  − � � √  − � z2

1 =

π

2

� √  − �

+ i ln iz  +

π

2

� � − � − � �� ��  � ��  −  − − ��  � i z

arccot(z ) = 21 i ln 1 arcsec(z ) = i ln

arccsc(z ) = −i ln

1

ln 1 +

+

�

π

2

 −

1+

z

2

1

− z1

2

+

1

z

i z

=  i  ln

1

1 z

2

i z

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

 − arcsin(z) z̸ = −1, +1

which has the same cut as arcsin;

z2 =

arctan(z ) = 21 i [ln (1 − iz ) − ln (1 +  iz )]

where (the square-root function has its cut along the neg- 4.1.1 Example proof ative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the cut between the sin(φ) =  z principal sheet of arcsin and other sheets; φ  =  arcsin(z ) Using the exponential definition of sine, one obtains arccos(z ) =

1

z  =

eφi

− e− 2i

φi

i z

+

π

2

=

2

6

5

Let

5.1.1

APPLICATIONS 

Application: finding the angle of a right triangle

ξ  =  e φi

Solving for  φ

z  =

ξ 

−

1

ξ

2i

2iz  =  ξ 

− ξ 1

− 2iz −  1ξ  = 0 ξ  − 2iξz − 1 = 0 ξ  =  iz ± 1 − z =  e φi  =  ln iz ± 1 − z φ  = −i ln iz ± 1 − z ξ 

2

√  � √  � � √  � 2

φi

2

2

(the positive branch is chosen) A right triangle.

φ  =  arcsin(z ) =

5 5.1

−i ln

� √  − � iz +

Applications

1

z2

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine, for example, it follows that

General solutions

  opposite Each of the trigonometric functions is periodic in the real θ  =  arcsin hypotenuse . part of itsargument, running through allits values twice in each interval of 2π. Sine and cosecant begin their period Often, the hypotenuse is unknown and would need to at 2πk   − π/2 (where  k  is an integer), finish it at 2πk  + be calculated before using arcsine or arccosine using the π/2, and then reverse themselves over 2π k  + π/2 to 2π k  + Pythagorean Theorem: a2 +  b 2 = h2 where h  is the 3π/2. Cosine and secant begin their period at 2π k , finish length of the hypotenuse. Arctangent comes in handy it at 2πk  + π, and then reverse themselves over 2π k  + π to in this situation, as the length of the hypotenuse is not 2πk  + 2π. Tangent begins its period at 2πk  − π/2, finishes needed. it at 2πk  + π/2, and then repeats it (forward) over 2π k  + π/2 to 2πk  + 3π/2. Cotangent begins its period at 2πk , finishes it at 2πk  + π, and then repeats it (forward) over θ  =  arctan opposite . adjacent 2πk  + π to 2πk  + 2π.

�

�

�

�

This periodicity is reflected in the general inverses where For example, suppose a roof drops 8 feet as it runs out k  is some integer: 20 feet. The roof makes an angle  θ  with the horizontal, where  θ  may be computed as follows: sin(y ) = x

⇔ y =  arcsin(x)+2πk or y =  π −arcsin(x)+2πk 8 opposite rise θ  =  arctan = arctan = arctan ≈ 21.8◦ . sin(y ) = x ⇔ y  = (−1) arcsin(x) + πk adjacent run 20 cos(y) = x ⇔ y  =  arccos(x)+2πk or y  = 2π−arccos(x)+2πk cos(y) = x ⇔ y  = ± arccos(x) + 2πk 5.2 In computer science and engineering tan(y ) =  x ⇔ y  =  arctan(x) + πk 5.2.1 Two-argument variant of arctangent cot(y ) = x ⇔ y  =  arccot(x) + πk )+2πkarticle: atan2 sec(y ) = x ⇔ y  =  arcsec(x)+2πk  or  y  = 2π−arcsec(xMain csc(y ) = x ⇔ y  =  arccsc(x)+2πk  or  y  =  π −arccsc(x)+2πk k

�

�

� �

� �

7 The two-argument atan2 function computes the arctangent of y / x  given y and x , but with a range of (−π, π]. In other words, atan2(y,  x ) is the angle between the positive  x -axis of a plane and the point ( x , y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower halfplane,  y  < 0). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.

computer implementation (due to the limited number of digits).[8] Similarly, arcsine is inaccurate for angles near −π/2 and π/2.

6

•  Inverse exsecant •  Inverse versine •  Inverse hyperbolic function •  List of integrals of inverse trigonometric functions • List of trigonometric identities •  Trigonometric function

In terms of the standard  arctan  function, that is with range of (−π/2, π/2), it can be expressed as follows:

atan2(y, x) =

arctan( xy ) arctan( xy ) + π arctan( xy ) − π

   −

x >  0

 ≥ 0  , x < 0

y

y < 0  , x < 0

π

y > 0  , x  = 0

2

π

y < 0  , x  = 0

2

undefined

y  = 0  , x  = 0

It also equals the principal value of the argument of the 7 complex number  x  +  iy. This function may also be defined using the tangent halfangle formulae as follows:

atan2(y, x) = 2 arctan

�

y

√ 

x2 + y 2 + x

�

provided that either  x  > 0 or  y  ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use. The above argument order ( y, x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention ( x , y) so some caution is warranted. These variations are detailed at atan2. 5.2.2

Arctangent function with location parameter

In many applications the solution y  of the equation  x = tan y  is to come as close as possible to a given value −∞ < η < ∞  . The adequate solution is produced by the parameter modified arctangent function

y  =  arctanη (x) := arctan(x) +  π

(x) . · rni η − arctan π

The function rni rounds to the nearest integer. 5.2.3

See also

Numerical accuracy

For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a

References

[1] For example Dörrie, Heinrich (1965).  Triumph der Mathematik . Trans. David Antin. Dover. p. 69.  ISBN 0-48661348-8. [2] Beach, Frederick Converse; Rines, George Edwin, eds. (1912). “Inverse trigonometric functions”.   The Americana: a universal reference library. 21. [3]   Cajori, Florian (1919).  A History of Mathematics  (2 ed.). New York, USA: The Macmillan Company. p. 272. [4]  Herschel, John Frederick William   (1813).   “On a remarkable Application of Cotes’s Theorem”.  Philosophical  Transactions . Royal Society, London. 103 (1): 8. [5] Korn, Grandino Arthur; Korn, Theresa M. (2000) [1961]. “21.2.−4. Inverse Trigonometric Functions”. Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review (3 ed.).

Mineola, New York, USA: Dover Publications, Inc. p. 811. ISBN 978-0-486-41147-7. [6] Bhatti, Sanaullah; Nawab-ud-Din; Ahmed, Bashir; Yousuf, S. M.; Taheem, Allah Bukhsh (1999). “Differentiation of Trigonometric, Logarithmic and Exponential Functions”. In Ellahi, Mohammad Maqbool; Dar, Karamat Hussain; Hussain, Faheem.  Calculus and Analytic Geometry (1 ed.). Lahore: Punjab Textbook Board. p. 140. [7] Borwein, Jonathan; Bailey, David; Gingersohn, Roland (2004).   Experimentation in Mathematics: Computational  Paths to Discovery (1 ed.). Wellesley, MA:A K Peters. p. 51. ISBN 1-56881-136-5. [8] Gade, Kenneth (2010).   “A non-singular horizontal position representation”   (PDF).   The Journal of Navigation. Cambridge University Press. 63 (3): 395–417. doi:10.1017/S0373463309990415.

8

8

8

External links

•  Weisstein, Eric W. “Inverse Trigonometric Functions”.  MathWorld .

•  Weisstein, Eric W. “Inverse Tangent”.  MathWorld .

EXTERNAL LINKS 

9

9

Text and image sources, contributors, and licenses

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9.2 •

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Images File:Arcsecant_Arccosecant.svg  Source:    https://upload.wikimedia.org/wikipedia/commons/5/56/Arcsecant_Arccosecant.svg   License:  CC BY-SA 3.0  Contributors:  Own work  Original artist:  Geek3  File:Arcsine_Arccosine.svg  Source:  https://upload.wikimedia.org/wikipedia/commons/b/b4/Arcsine_Arccosine.svg  License:  CC SA 3.0 Contributors:  Own work  Original artist:  Geek3

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Gudermannian function

Graph of the Gudermannian function

Graph of the inverse Gudermannian function

The  Gudermannian function , named after  Christoph Gudermann   (1798–1852), relates the   circular functions and  hyperbolic functions  without explicitly using complex numbers.

1.3

x

−1

gd

1

 x

1

0

cosh t

∫ 

1 cos t

 dt

− π /2  < x < π /2

 1 + sin x  1  1 + sin x   = 2 ln  1 − sin x  =  ln  cos x   � � =  ln |tan x + sec x| =  ln tan π + x 

 dt.

1 4

1 2

=  artanh(sin x) =  arsinh(tan x) =  arcoth(csc x) =  arcsch(cot x)

Properties

1.1

x  =

�  0

It is defined for all  x  by[1][2][3]

gd x  =

Inverse

=  sgn (x) arcosh(sec x) =  sgn (x) arsech(cos x) = − i gd(ix)

Alternative definitions

(See inverse hyperbolic functions.)

gd x  =  arcsin (tanh x) =  arctan(sinh x) =  arccsc(coth x)

=  sgn (x) ·  arccos (sech x) =  sgn (x) ·  arcsec(cosh x) 1.4 1

� � �� = 2 arctan tanh x

sinh(gd−1 x) =  tan x;   csch(gd−1 x) =  cot x;

2

= 2 arctan(ex ) − 21 π.

cosh(gd−1 x) =  sec x;   sech(gd−1 x) =  cos x; tanh(gd−1 x) =  sin x;   coth(gd−1 x) =  csc x.

Some related formula, such as arccot (csch x)  , doesn't quite work as definition. (See inverse trigonometric functions.)

1.2

1.5

csc(gd x) =  coth x;

cos(gd x) =  sech x;

sec(gd x) =  cosh x;

tan(gd x) =  sinh x;

cot(gd x) =  csch x;

Derivatives

d  gd x  =  sech x; dx

Some identities

sin(gd x) =  tanh x;

Some identities

2

d gd−1 x  =  sec x. dx

History

The function was introduced by  Johann Heinrich Lambert in the 1760s at the same time as the  hyperbolic functions. He called it the “transcendent angle,” and it went

tan( 12  gd x) =  tanh( 12 x). 1

2

5

by various names until 1862 when  Arthur Cayley   suggested it be given its current name as a tribute to Gudermann’s work in the 1830s on the theory of special functions.[4] Gudermann had published articles in  Crelle’s  Journal  that were collected in  Theorie der potenzial- oder  cyklisch-hyperbolischen Functionen  (1833), a book which expounded  sinh  and  cosh  to a wide audience (under the guises of  Sin  and  Cos  ). The notation gd   was introduced by Cayley [5] where he starts by calling   gd. u  the inverse of the  integral of the secant function:

φ

u  =

�  0

1

1

4

2

� � �� sec t dt  =  ln tan π  + φ

and then derives “the definition” of the transcendent:

5

REFERENCES 

References

[1] Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W., eds. (2010),  NIST Handbook of Mathematical Functions , Cambridge University Press. Section 4.23(viii). [2] CRC   Handbook of Mathematical Sciences   5th ed. pp. 323–325 [3]  Weisstein, Eric W. “Gudermannian”.  MathWorld . [4] George F. Becker, C. E. Van Orstrand.  Hyperbolic functions.  Read Books, 1931. Page xlix. Scanned copy available at archive.org [5]   Cayley, A.   (1862). “On the transcendent gd. u”. 19–21. Philosophical Magazine (4th ser.). 24: doi:10.1080/14786446208643307 (inactive 201701-16). [6] Osborne, P (2013), The Mercator projections , p74

gd u  =  i −1 ln tan

1

1

4

2

� � π + ui��

observing immediately that it is a real function of  u .

3

Applications • The angle of parallelism  function in hyperbolic geometry is defined by

1 2

π −  gd x

• On a  Mercator projection a line of constant latitude is parallel to the equator (on the projection) and is displaced by an amount proportional to the inverse Gudermannian of the latitude.

• The Gudermannian (with a complex argument) may be used in the definition of the transverse Mercator projection.[6]

•  The Gudermannian appears in a non-periodic solution of the inverted pendulum.[7]

• The Gudermannian also appears in a moving mirror solution of the dynamical Casimir effect.[8]

4

See also •  Hyperbolic secant distribution •   Mercator projection •   Tangent half-angle formula •   Tractrix •   Trigonometric identity

[7] John S. Robertson (1997). “Gudermann and the Simple Pendulum”. The College Mathematics Journal . 28  (4): 271–276. JSTOR 2687148. Review. [8] Good, Michael R. R.; Anderson, Paul R.; Evans, Charles R. (2013). “Time dependence of particle creation from accelerating mirrors”. Physical Review D . 88 (2): 025023. arXiv:1303.6756 .   Bibcode:2013PhRvD..88b5023G. doi:10.1103/PhysRevD.88.025023.

3

6

Text and image sources, contributors, and licenses

6.1 •

6.2 •

•

•

6.3 •

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Error function no greater than  x . This is true for any random variable with distribution (0, 12 )  ; but the application to error variables is how the error function got its name.

1.00

 N 

0.75

0.50

The previous paragraph can be generalized to any variance: given a variable (such as an unbiased error vari(0, σ 2 )  , evaluating the error function at able) ε

0.25

       )      x        (

        f0.00       r       e

−0.25

erf

−0.50

−1.00 − 2

−1

0

1

2

3

x

Plot of the error function

2

In   mathematics, the   error function   (also called the Gauss error function) i s a   special function   (nonelementary) of sigmoid shape that occurs in  probability, statistics, and partial differential equations describing diffusion. It is defined as: [1][2]

erf(x) =

=

x

1

∫  √  ∫  √ 

π −x x

2

π

2.1

1 √ 

2

describes the probability of  ε  falling in the

Derived and related functions Complementary error function

The  complementary error function , denoted erfc , is defined as

2

− erf(x) ∞ 2 e− = √  π

erfc(x) = 1

e−t dt

∫ 

2

e−t dt.

t2

dt

x

0

2

=  e −x erfcx(x),

In statistics, for nonnegative values of x , the error function has the following interpretation: for a random variable  X  that is normally distributed with mean 0 and variance 12 , erf( x ) describes the probability of  X  falling in the range [− x , x ].

1

x σ

range [− x , x ].[3] This is used in statistics to predict behavior of any sample with respect to the population mean. This usage is similar to the Q-function, which in fact can be written in terms of the error function.

−0.75

−3

�  · ∼ N �

which also defines erfcx , the   scaled complementary error function[4] (which can be used instead of erfc to avoid   arithmetic underflow[4][5] ). Another form of erfc(x) for non-negative x is known as Craig’s formula:[6]

The name 'error function'

2

| ≥ 0) = π

erfc(x x

π/2

x2

∫  �− � exp

0

sin2 θ

dθ.

The error function is used in measurement theory (using probability and statistics), and its use in other branches of 2.2 Imaginary error function mathematics is typically unrelated to the characterization of measurement errors. The  imaginary error function , denoted  erfi, is defined In statistics, it is common to have a variable Y   and its as ˆ . The error is then defined as unbiased estimator Y  ˆ ε = Y  Y  . This makes the error a normally distributed random variable with mean 0 (because the esti- erfi(x) = i erf(ix) mator is unbiased) and some variance  σ 2 ; this is written x 2 (0, σ 2 )  . For the case where  σ 2 = 12  , i.e. an as  ε = et dt π 0 unbiased error variable  ε (0, 12 )  , erf( x ) describes the probability of the error  ε  falling in the range [− x ,  x ]; 2 x e D (x), = in other words, the probability that the absolute error is π

−

 ∼ N 

−

∫  √ 

 ∼ N 

√ 

1

2

2

2

3

PROPERTIES 

where  D ( x ) is the  Dawson function (which can be used instead of erfi to avoid arithmetic overflow[4] ). Despite the name “imaginary error function”, erfi(x)  is real when  x  is real. When the error function is evaluated for arbitrary complex arguments z, the resulting  complex error function  is usually discussed in scaled form as the  Faddeeva function:

2

w(z ) =  e −z erfc( iz ) =  erfcx( iz ).

−

−

erf( z)

−

 −

The property erf( z ) = erf(z )  means that the error function is an odd function. This directly results from the fact that the integrand e −t is an even function. 2

2.3

Cumulative distribution function

For any complex number  z :

The error function is related to the cumulative distribution Φ , the integral of the standard normal distribution, by [2] erf(z ) =  erf (z ) where  z  is the complex conjugate of  z .

� √  �

1  1 1 Φ(x) = +  erf x/ 2 =  erfc 2 2 2

3

Properties

Plots in the complex plane

�− √  �

x/ 2 .

The integrand  ƒ  = exp(− z2 ) and  ƒ  = erf( z) are shown in the complex  z -plane in figures 2 and 3. Level of Im( ƒ ) = 0 is shown with a thick green line. Negative integer values of Im( ƒ ) are shown with thick red lines. Positive integer values of Im( f ) are shown with thick blue lines. Intermediate levels of Im( ƒ ) = constant are shown with thin green lines. Intermediate levels of Re( ƒ ) = constant are shown with thin red lines for negative values and with thin blue lines for positive values. The error function at +∞ is exactly 1 (see  Gaussian integral). At the real axis, erf( z) approaches unity at  z  → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ± i ∞.

3.1

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its  Taylor expansion always converges. The defining integral cannot be evaluated in closed form in terms of  elementary functions, but by expanding the integrand e− z into its Maclaurin series   and integrating term by term, one obtains the error function’s Maclaurin series as: 2

erf(z ) =

exp(− z2 )

∞

( 1)n z 2n+1 2 = π n!(2n + 1) π n=0

∑− √  2

� √  − z

z3

3

+

z5

7

9

z z  −  − ·· + 10 42 216

Integrand which holds for every  complex number  z . The denominator terms are sequence A007680 in the OEIS.

3.3

3

Bürmann series 

3.3

For iterative calculation of the above series, the following alternative formulation may be useful:

� ∏− ∑ √ 

erf(z ) =

n

∞

2

π

z

n=0

k=1

(2k 1)z 2 k (2k + 1)

−

�

=

An expansion,[8] which converges more rapidly for all real values of  x  than a Taylor expansion, is obtained by using [9] n zHans Heinrich z 2 Bürmann's theorem:

∞

2

∑ √  π

n=0

Bürmann series

∏−

2n + 1

k=1

erf(x)

2

−(2k−1)z because k(2k+1) expresses the multiplier to turn the th

k

th

k  term into the (k  + 1) term (considering z  as the first term).

=

2

√ π  sgn (x)

√  − � − e−x

1

2

1

1 (1 12

erfi(z ) =

∞

2

∑ √  π

n=0

z 2n+1 2 = n!(2n + 1) π

z +

z3

3

+

z5

10

 +

7 z=

42

2

√ � − � √  ∑ ···

9 (x) √ πz sgn

 +

216

+

e−x

1

π

2

2

+

∞

k=1

2

ck e−kx

)

�

 − ·

The derivative of the error function follows immediately from its definition:

d 2 −z e .  erf (z ) = dz π

787 (1 − e− − 276480

.

2

±

√  − � √ 

2

erf(x)

≈ √ π  sgn (x)

3.4

Inverse functions

1

2

e−x

31 −x e + 2 200 π

2

−

341 −2x e 8000

From this, the derivative of the imaginary error function is also immediate:

d 2 z e .  erfi(z ) = dz π

√ 

2

An   antiderivative   of the error function, obtainable by integration by parts, is

2

e−z z erf(z ) + . π

√ 

Inverse error function

Given complex number  z , there is not a  unique  complex An antiderivative of the imaginary error function, also obnumber  w  satisfying erf(w) =  z  , so a true inverse functainable by integration by parts, is tion would be multivalued. However, for −1 < x  < 1, there is a unique real   number denoted erf−1 (x) satisfying erf erf−1 (x) =  x  . z e z erfi(z ) . The  inverse error function  is usually defined with doπ main (−1,1), and it is restricted to this domain in many computer algebra systems. However, it can be extended Higher order derivatives are given by to the disk | z| < 1 of the complex plane, using the Maclaurin series k−1 k−1 2( 1) 2 d e−z , k  = 1, 2, . . . H k−1 (z )e−z = erf(k) (z ) = π π dz k−1 ∞ 2k+1 ck π − 1 z , erf (z ) = 2k + 1 2 where  H  are the physicists’ Hermite polynomials.[7]

(

2

− √ 

−√ 

2

√ 

)

By keeping only the first two coefficients and choosing 31 341 c1 = 200 and c 2 = 8000  , the resulting approximation shows its largest relative error at  x  = 1.3796  , where it is less than  3.6127 10−3 :

Derivative and integral

√ 

x2 2

x2 3

which holds for every complex number  z .

3.2

x2

5 (1 − e− − 896

The imaginary error function has a very similar Maclaurin series, which is:

� √ 

7 − e− ) − 480 (1 − e−

� �



2

∑

k=0

� √  �

2

�

.

4

4

APPROXIMATION WITH ELEMENTARY FUNCTIONS 

�

k 1

ck =

−

∑

m=0

2

RN (x) =  O x1−2N e−x

where  c 0  = 1 and

�

cm ck−1−m = (m + 1)(2m + 1)

�

as  x

→ ∞.

Indeed, the exact value of the remainder is  7  127  4369  34807 , , ,... . 1, 1, , 6 90 2520 16200

�

So we have the series expansion (note that common fac( 1)N  1−2N  (2N )! ∞ −2N  −t RN (x) := t e dt, 2 tors have been canceled from numerators and denominaN ! π x tors): which follows easily by induction, writing e−t

−√ 

∫ 

2

2

erf−1 (z ) =

1 2

� � 2

′

=

−1 −t and5integrating by parts. πt4) 9 e   34807 π  7 π 2 5  127 π3 7   4369(2 π z  + z + z + z + z + z 11 + . 12 480 40320 5806080 182476800 For large enough values of x, only the first few terms of

√  �

π

−

3

� ···

(After cancellation the numerator/denominator fractions this asymptotic expansion are needed to obtain a good are entries   A092676/   A132467 in the OEIS; with- approximation of erfc( x ) (while for not too large values of  x  note that the above Taylor expansion at 0 provides a out cancellation the numerator terms are given in entry A002067.) Note that the error function’s value at ±∞ is very fast convergence). equal to ±1. For | z| < 1, we have erf erf−1 (z ) =  z  .

(



3.6

The  inverse complementary error function  is defined as erfc−1 (1

Continued fraction expansion

A continued fraction expansion of the complementary error function is:[11]

− z) = erf−1(z).

For real x , there is a unique  real  number erfi−1 (x)  satisfying erfi erfi−1 (x) = x   . The  inverse imaginary error function  is defined as erfi −1 (x)  . [10]

(

erfc(z ) =



√ zπ e−

1

z2

a1

z2 +

For any real  x , Newton’s method can be used to compute erfi−1 (x)  , and for 1 x 1  , the following Maclaurin series converges:

z2 +

− ≤ ≤

erfi−1 (z ) =

∑ − � √  � ∞

( 1)k ck 2k + 1

k=0

π

2

z

erf

� ∑ √  − ∞

· · ···

1 3 5 (2n ( 1)n (2x2 )n n=1

− 1)

�

where (2n   – 1)!! is the double factorial: the product of all odd numbers up to (2n  – 1). This series diverges for every finite  x , and its meaning as asymptotic expansion is that, for any N  N one has

 ∈ 2

e−x erfc(x) = x π

N  1

−

∑− √  n=0

( 1)n

1+

···

−

(2n 1)!! + RN (x) (2x2 )n

where the remainder, in Landau notation, is

∞ erf (ax + b) = exp 2πd 2 −∞

� √  − � ∫  b

ac

1 + 2 a2 d2

4

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real  x  is 2

a3

,

Asymptotic expansion

e−x erfc(x) = 1+ x π

2

Integral of error function with Gaussian density function

2k+1

where  ck  is defined as above.

3.5

m  .

a2

1+

3.7

am  =

√ 

�−

(x + c)2 dx, 2d2

�

Approximation with elementary functions

Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose ∞ the fastest approximation suitable for a given e−x 1)!! n (2nof increasing application. accuracy, they are: , = ( In1)order 2 n (2x ) x π 2

√ 

∑−

−

n=0

erf(x)

≈

1

1 , (1+a1 x+a2 x2 +a3 x3 +a4 x4 )4 −4

(maximum error: 5×10 )

x

≥

− 0

where a1  = 0.278393,  a2  = 0.230389,  a3  = 0.000972,  a4 = 0.078108 erf(x) 1 , 1+ px

x2

≈ 1 − (a1t + a2t2 + a3t3)e− , t = x ≥ 0 (maximum error: 2.5×10 ) −5

a, b , c,

5

where  p  = 0.47047,  a 1  = 0.3480242,  a 2  = −0.0958798, a3  = 0.7478556 erfc(x)

≈

erf(x)

−

1

1 (1+a1 x+a2 x2 +···+a6 x6 )16 ,

≥

x

(maximum error: 3×10−7 )

≈

1

2

−

(a1 t + a2 t2 + 1 1+ px

t =

···

2e

β  1 −βx e , β 

π

2

x

≥ 0, β > 1,

where the parameter  β  can be picked to minimize error on the desired interval of approximation.

0

where a1   = 0.0705230784, a2   = 0.0422820123, a3 = 0.0092705272, a4 = 0.0001520143, a5 = 0.0002765672, a6  = 0.0000430638 erf(x) a5 t5 )e−x ,

≥

�  √  −

5

Numerical approximations

Over the complete range of values, there is an approximation with a maximal error of  1.2 10−7 , as follows:[15]

×

+

(maximum error:

1.5×10−7 )

erf(x) =

where p   = 0.3275911, a1   = 0.254829592, a2 = −0.284496736, a 3  = 1.421413741,  a 4  = −1.453152027, a5  = 1.061405429 All of these approximations are valid for x   ≥ 0. To use these approximations for negative x , use the fact that erf(x) is an odd function, so erf( x ) = −erf(− x ).

�− 1

τ 

 − 1

τ 

forx

≥0

forx <  0

with

· −x2 − 1.26551223 + 1.00002368t + 0.37409196t2 + 0.09 − 0.18628806t4 + 0.27886807t5 − 1.13520398t6 + 1.48851587t − 0.82215223t8 + 0.17087277t9

τ  =  t exp

(



Another approximation is given by and

erf(x)

�  � − −

≈ sgn(x)

1

exp

4

+ ax2 x2 π 1 + ax2

�

t  =

||

Also, over the complete range of values, the following simple approximation holds for x erfc(x)  , with a maximal error of  6.3 10−4 :

where

a  =

1 . 1 + 0 .5 x

8(π 3π (4

×

− 3)  ≈ 0.140012. − π)

x erfc(x) = 1.3693x exp

This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the error is less than 0.00035 for all x . Using the alternate value a ≈ 0.147 reduces the maximum error to about 0.00012.[12] This approximation can also be inverted to calculate the inverse error function:

6

(−

0.8072(x + 0.6388)2

Applications

When the results of a series of measurements are described by a normal distribution with standard deviation

σ  and expected value 0, then erf

erf−1 (x)

  � �

≈ sgn(x)

2 πa

 +

ln(1

− x2 ) 2

2

�−

≤ 21 e−2 1 erfc(x) ≈ e− 6 erfc(x)

 1 e−x , x > 0 + e−x 2  1 x >  0 . + e− x , 2

x2

x2

2

4 3

≤

2

� � σ

a √ 

2

is the probabil-

ity that the error of a single measurement lies between positive (1 x a2.) This is useful, for example, in ln(1 x−2a) and +a2, for ln . a digital communication  + rate of a determining πa the bit error 2 system.

−

Exponential bounds and a pure exponential approximation for the complementary error function are given by [13]



� −

−

�

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function. The error function and its approximations can be used to estimate results that hold   with high probability. Given random variable  X  Norm[µ, σ ]  and constant  L < µ  :

 ∼

2

A single-term lower bound is [14]

 ≤

Pr[X 

1 1 L] = +  erf 2 2

� √ − � ≈ �− � − � � L

µ

2σ

A exp

B

L

µ

σ

2

6

7

where A  and  B  are certain numeric constants. If L  is sufficiently far from the mean, i.e. µ L σ ln k  , then:

− ≥ √ 

7

1 2 ,x erf(x) =  sgn (x)P  2

� �

sgn(x) 1 2 γ  ,x = π 2

√ 

.

sgn(x) is the sign function.

 ≤ L] ≤ A exp(−B ln k) = kA so the probability goes to 0 as  k → ∞ . Pr[X 

� �

RELATED FUNCTIONS 

B

7.1

Generalized error functions n=5

Related functions

n=4

3

n=3  2

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by software languages, as they differ only by scaling and translation. Indeed,

n=2

1

n=1

      )     x       (

    n

0

      E −1

− 2

Φ(x) =

x −t 1 1 e 2 dt  = 1 +  erf 2 2π −∞ 2

∫ 

√ 

�

� √  �� x

2

1 =  erfc 2

�− √  � −3

x

−3

2

− 2

−1

0

1

2

3

x

or rearranged for erf and erfc:

� √  � − �− √  � � − � √  ��

erf(x) = 2Φ x 2 erfc(x) = 2Φ

π

1

x 2 =2 1

Φ x 2

.

Consequently, the error function is also closely related to the   Q-function, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as

Q(x) =

1 2

 −  12  erf

� √  � x

2

1 =  erfc 2

� √  � x

2

Graph of generalised error functions  E n(x):  √   grey curve:  E1 (x) = (1 − e −x )/  red curve:  E2 (x) = erf(x)  green curve:  E3 (x) blue curve:  E4 (x)  gold curve:  E5 (x).

.

The inverse of  Φ  is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

Some authors discuss the more general functions:

n! E n (x) = π

x

∫  √ 

n! e−t dt  =

∞

xnp+1 . ( 1) (np + 1) p!

∑ √  −

n

π

0

 p

 p=0

Notable cases are:

• E 0( x ) is a straight line through the origin: E 0(x) = x e π

√ 

• E 2( x ) is the error function, erf( x ). After division by n!, all the En  for odd n look similar (but

√  √  not identical) to each other. Similarly, the  En  for even  n probit( p) = Φ −1 ( p) = 2 erf−1 (2 p−1) = − 2 erfc−1 (2 p).

look similar (but not identical) to each other after a simple division by  n !. All generalised error functions for n  > 0 The standard normal cdf is used more often in probability look similar on the positive x  side of the graph. and statistics, and the error function is used more often in other branches of mathematics. These generalised functions can equivalently be expressed for x    > 0 using the   Gamma function and The error function is a special case of the   Mittag-Leffler incomplete Gamma function: function, and can also be expressed as a  confluent hypergeometric function (Kummer’s function):

E n (x) = erf(x) =

2x √  π

F 1 1

�

�

1  3 , 2 , x2 . 2

−

1

√ π Γ(n)

� � � − � �� Γ

1

n

1

n

, xn

,

x >  0.

Therefore, we can define the error function in terms of the incomplete Gamma function:

It has a simple expression in terms of the Fresnel integral. In terms of the regularized Gamma function P  and the incomplete gamma function,

Γ

erf(x) = 1

1

− √ π Γ

� �

1 2 ,x . 2

7

7.2

Iterated integrals of the complementary error function

• Go: Provides math.Erf() and math.Erfc() for float64

The iterated integrals of the complementary error function are defined by

•   Google search: Google’s search also acts as a cal-

n

i erfc(z ) =

∞

∫ 

arguments.

culator and will evaluate “erf(...)" and “erfc(...)" for real arguments. An erf package[18] exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

•   Haskell:

n 1

i − erfc(ζ ) dζ.

z

They have the power series

• IDL: provides both erf and erfc for real and complex ∞

arguments.

( z )j in erfc(z ) = , n−j j !Γ 1 + n−j 2 2 j =0

−

∑

(

Apache commons-math[19] provides implementations of erf and erfc for real arguments.

•   Java:



from which follow the symmetry properties

•   Julia: Includes erf and erfc for real and complex arguments. Also has erfi for calculating  i erf(ix)

m

i2m erfc( z ) =

−

−i2

m

∑

erfc(z )+

q =0

z 2q

22(m−q)−1 (2q )!(m

Maple: Maple implements both erf and erfc for real − q )•!   and complex arguments.

and

•   MathCAD provides both erf(x) and erfc(x) for real arguments.

m

∑

i2m+1 erfc( z ) =  i 2m+1 erfc(z )+

−

8

q =0

z 2q+1

•   Mathematica: erf is implemented as Erf and Erfc in − q )! .

22(m−q)−1 (2q  + 1)!(m

Implementations

Mathematica for real and complex arguments, which are also available in Wolfram Alpha.

•   Matlab   provides both erf and erfc for real arguments, also via W. J. Cody’s algorithm.[20]

• C: C99 provides the functions double erf(double x)

and double erfc(double x) in the header math.h. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. For complex double arguments, the function names cerf and cerfc are “reserved for future use"; the missing implementation is provided by the open-source project   libcerf, which is based on the Faddeeva package.

• C++: C++11 provides erf() and erfc() in the header

cmath. Both functions are overloaded to accept arguments of type float, double, and long double. For complex, the  Faddeeva package  provides a C++ complex implementation.

• D: A D package

[16]

exists providing efficient and accurate implementations of complex error functions, along with Dawson, Faddeeva, and Voigt functions.

•   Excel: Microsoft Excel provides the erf, and the erfc functions, nonetheless both inverse functions are not in the current library.[17]

•   Fortran:

The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED functions to calculate the error function and its complement for real arguments.   Fortran 77  implementations are available in SLATEC.

•   Maxima provides both erf and erfc for real and complex arguments.

•   PARI/GP: provides erfc for real and complex arguments, via tanh-sinh quadrature plus special cases.

•   Perl:

erf (for real arguments, using Cody’s algorithm[20] ) is implemented in the Perl module Math::SpecFun

•   Python:

Included since version 2.7 as math.erf() and math.erfc() for real arguments. For previous versions or for complex arguments,   SciPy includes implementations of erf, erfc, erfi, and related functions for complex arguments in scipy.special. [21] A complex-argument erf is also in the  arbitrary-precision arithmetic  mpmath library as mpmath.erf() [22]

• R: “The so-called 'error function'"

is not provided directly, but is detailed as an example of the  normal cumulative distribution function (?pnorm), which is based on W. J. Cody’s rational Chebyshev approximation algorithm.[20]

•   Ruby: Provides Math.erf() and Math.erfc() for real arguments.

8

11

9

See also

9.1

[10] Bergsma, Wicher.   “On a new correlation coefficient, its orthogonal decomposition and associated tests of independence” (PDF).

Related functions

•  Gaussian integral, over the whole real line •   Gaussian function, derivative •   Dawson function, renormalized imaginary

[11] Cuyt, Annie A. M.; Petersen, Vigdis B.; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). Handbook of Continued Fractions for Special Functions . Springer-Verlag. ISBN 978-1-4020-6948-2.

error

function

•  Goodwin–Staton integral 9.2

In probability

•  Normal distribution •   Normal cumulative distribution function, a scaled and shifted form of error function

•   Probit, the inverse or quantile function of the normal CDF

•   Q-function, the tail probability of the normal distribution

10

FURTHER READING 

References

[12] Winitzki, Sergei (6 February 2008).   “A handy approximation for the error function and its inverse” (PDF). Retrieved 2011-10-03. [13]   Chiani, M., Dardari, D., Simon, M.K. (2003). New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels . IEEE Transactions on Wireless Communications, 4(2), 840– 845, doi=10.1109/TWC.2003.814350. [14] Chang, Seok-Ho; Cosman, Pamela C.; Milstein, Laurence B. (November 2011). “Chernoff-Type Bounds IEEE Transacfor the Gaussian Error Function”. tions on Communications . 59 (11): 2939–2944. doi:10.1109/TCOMM.2011.072011.100049. [15] Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 0-521-43064-X), 1992, page 214, Cambridge University Press. [16]  DlangScience/libcerf, A package for use with the D Programming language.

[1] Andrews, Larry C.;  Special functions of mathematics for  engineers 

[17] These results can however be obtained using the NormSInv function as follows: erf_inverse(p) = NormSInv((1 - p)/2)/SQRT(2); erfc_inverse(p) = NormSInv(p/2)/SQRT(2). See .

[2] Greene, William H.; Econometric Analysis  (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11

[18]   http://hackage.haskell.org/package/erf

[3] Van Zeghbroeck, Bart;   Principles of Semiconductor Devices , University of Colorado, 2011. [4] Cody, W. J. (March 1993),   “Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers”   (PDF), ACM Trans. Math. Softw., 19 (1): 22–32, doi:10.1145/151271.151273 [5] Zaghloul, M. R. (March 1, 2007),   “On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand”,   Monthly Notices of  the Royal Astronomical Society, 375   (3): 1043–1048, doi:10.1111/j.1365-2966.2006.11377.x [6]   John W. Craig, A new, simple and exact result for calculating the probability oferror for two-dimensional signal constellations , Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575. [7]  Wolfram MathWorld [8] H. M. Schöpf and P. H. Supancic, “On Bürmann’s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion,” The Mathematica Journal, 2014. doi:10.3888/tmj.16–11.Schöpf, Supancic [9] E. W. Weisstein. “Bürmann’s Theorem” from Wolfram MathWorld—A Wolfram Web Resource./ E. W. Weisstein

[19]   Commons Math: The Apache Commons Mathematics Library [20] Cody, William J. (1969). “Rational Chebyshev Approximations for the Error Function”  (PDF). Math. Comp. 23   (107): 631–637. doi:10.1090/S0025-5718-19690247736-4. [21]  Error Function and Fresnel Integrals, SciPy v0.13.0 Reference Guide. [22] R Development Core Team (25 February 2011), R: The Normal Distribution

11

Further reading

•   Abramowitz,

Milton;   Stegun, Irene Ann, eds. (1983) [June 1964].   “Chapter 7”.   Handbook of  Mathematical Functions with Formulas, Graphs, and  Mathematical Tables . Applied Mathematics Series. 55  (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 297. ISBN 978-0-486-61272-0.   LCCN 64-60036. MR 0167642. LCCN 65-12253.

9

•   Press, William H.; Teukolsky, Saul A.; Vetterling,

William T.; Flannery, Brian P. (2007), “Section 6.2. Incomplete Gamma Function and Error Function”, Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

•  Temme, Nico M. (2010),   “Error Functions, Dawson’s and Fresnel Integrals”, in   Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W.,  NIST Handbook of Mathematical Functions , Cambridge University Press,   ISBN 9780521192255, MR 2723248

12

External links

•  MathWorld – Erf • A Table of Integrals of the Error Functions

10

13

13

Text and image sources, contributors, and licenses

13.1 •

•

•

•

•

•

Images

  File:ComplexErf.jpg Source:  https://upload.wikimedia.org/wikipedia/commons/0/00/ComplexErf.jpg License:   CCBY 3.0 Contributors:  This mathematical image was created with Mathematica Original artist:   Domitori   File:ComplexEx2.jpg  Source:   https://upload.wikimedia.org/wikipedia/commons/1/18/ComplexEx2.jpg License:  CC BY-SA 3.0  Contributors:  This mathematical image was created with Mathematica Original artist:   Domitori  File:Error_Function.svg   Source:   https://upload.wikimedia.org/wikipedia/commons/2/2f/Error_Function.svg  License:   Public domain Contributors:  self-made, Inkscape Original artist:   Inductiveload File:Error_Function_Generalised.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/0/02/Error_Function_Generalised. svg License:  Public domain  Contributors:  self-made, Inkscape and Mathematica Original artist:   Inductiveload   File:Mplwp_erf_inv.svg Source:  https://upload.wikimedia.org/wikipedia/commons/e/ec/Mplwp_erf_inv.svg License:  CC BY 3.0  Contributors:  Own work Original artist:  Geek3  File:OEISicon_light.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/d/d8/OEISicon_light.svg   License:  Public domain   Contributors:  Own work   Original artist:     Watchduck (a.k.a. Tilman Piesk)

13.3 •

Text

  Error function   Source:   https://en.wikipedia.org/wiki/Error_function?oldid=781069539  Contributors:   AxelBoldt, Bryan Derksen, Tarquin, Michael Hardy, Chinju, Cyp, Stevenj, BenKovitz, Gutza, W ilke, Finlay McWalter, LaurentPerrinet, Chris Roy, Tualha, Saforrest, Wile E. Heresiarch, Tea2min, Giftlite, Lethe, Jorge Stolfi, Sietse, Nayuki, Vivero~enwiki, Alberto da Calvairate~enwiki, Nickptar, Urhixidur, Fintor, Askewchan, Chepry, Rich Farmbrough, MuDavid, Bender235, Zaslav, Kwamikagami, Shanes, Rbj, Larryv, Danski14, Sligocki, PAR, RJFJR, Oleg Alexandrov, Waabu, Linas, Gisling, Karam.Anthony.K, Qwertyus, Rjwilmsi, Strait, Salix alba, R.e.b., Erkcan, Alejo2083, FlaBot, Ausinha, Scythe33, Chet Gray, Chobot, Meawoppl, Peterl, Dstrozzi, YurikBot, Nbrouard, Pnrj, Crasshopper, Dbfirs, Entropeneur, NormDor, Arthur Rubin, Pred, Zvika, SmackBot, BahramH, GoOdCoNtEnT, Anachronist, Bh3u4m, Oli Filth, Papa November, Nbarth, Cybercobra, Cubbi, Mwtoews, PSeibert~enwiki, Sbmehta, Texas Dervish, Domitori, GeordieMcBain, CRGreathouse, Doomed Rasher, Thijs!bot, Ϙ, Headbomb, AllUltima, Lklundin, Xypron, Catslash, Albmont, QrczakMK, Baccyak4H, Beabroad, KConWiki, Tbleher, Sullivan.t.j, Falcor84, Dima373, Ricardogpn, Pedrito, Uncle Dick, Leaflet, Liangent, Salih, Stan J Klimas, KohanX, Theowoll, Epistemenical, DonAndre, Sheliak, X!, VolkovBot, PMajer, TXiKiBoT, Anonymous Dissident, Stafusa, Aaron Rotenberg, Jamelan, Flyer22 Reborn, JackSchmidt, Svick, AlanUS, Melcombe, Headlessplatter, Crgato, ClueBot, UKoch, Njasloane, P. M. Sakkas, IMneme, Pkeastman, Qwfp, DumZiBoT, AlexFekken, Addbot, Michele.allegra, Fgnievinski, Favonian, Jasper Deng, Luckas-bot, Yobot, Wateenellende, AnomieBOT, Erel Segal, Joule36e5, Hegpetz, Geek1337~enwiki, J acobakkerboom, Rubenstreb, ChrisKuklewicz, MathHisSci, Ichbin-dcw, SkyMachine, Duoduoduo, Olawlor, Thompson2212, Jxwx, John of Reading, Gz33, Zueignung, ClueBot NG, BarrelProof, Cntras, Dru of Id, Hikenstuff, Helpful Pixie Bot, Bgde, BG19bot, ServiceAT, Nsda, Hypnotoad33, Brad7777, Jagan seshadri, ChrisGualtieri, Qsq, Kondormari, Katterjohn, EmileContal, Kogge, Pscrape, Robdonne, Mchiani, Rafikmath15, DoctorTerrella, AntoineDeRivarol, Absolutelypuremilk, KasparBot, Boehm, Cdserio99, Dmt137, Bender the Bot, Deacon Vorbis and Anonymous: 191

13.2 •

TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

Content license

Creative Commons Attribution-Share Alike 3.0

Generalised logistic function

A=0, K=1, B=3, Q=ν=0.5, M=0, C=1

Effect of varying parameter B. A = 0, all other parameters are 1.

Effect of varying parameter A. All other parameters are 1.

The generalised logistic function  or  curve, also known as   Richards’ curve, originally developed for growth modelling, is an extension of the logistic or sigmoid functions, allowing for more flexible S-shaped curves:

Y  (t) = A +

Effect of varying parameter C. A = 0, all other parameters are 1.

K  − A (C  + Qe Bt )1/ν 

•

K   : the upper asymptote. If A = 0  then K   is called the carrying capacity;

•

B  : the growth rate;

•

ν > 0  : affects near which asymptote maximum growth occurs.

−

where Y  = weight, height, size etc., and t = time. It has five parameters: •

A : the lower asymptote; 1

2

1

GENERALISED LOGISTIC DIFFERENTIAL EQUATION 

Effect of varying parameter K. A = 0, all other parameters are 1.

Effect of varying parameter  ν  . A = 0, all other parameters are 1.

this representation simplifies the setting of both a starting time and the value of Y at that time. The logistic, with maximum growth rate at time  M  , is the case where Q = ν  = 1.

1

Generalised logistic differential equation

A particular case of the generalised logistic function is:

Y  (t) =

K  (t−t0 ) )1/ν 

−αν 

(1 + Qe

which is the solution of the so-called Richards’ differential equation (RDE): Effect of varying parameter Q. A = 0, all other parameters are 1. •

Q : is related to the value Y (0)

•

C   : typically takes a value of 1.

ν 

� � ��

′

Y  K 

Y  (t) = α 1 −

Y 

with initial condition

The equation can also be written:

Y  (t0 ) = Y 0 Y  (t) = A +

K  − A (C  + e B (t M ) )1/ν  −

−

where M   can be thought of a starting time, t0  (at which −A Y  (t0 ) = A + (C K  ) +1)1/ ν

Including both Q and M  can be convenient:

where

Q=

1+

−

ν 

� � K  Y 0

provided that ν > 0 and α > 0.

Y  (t) = A +

K  − A (C  + Qe B (t M ) )1/ν  −

−

The classical logistic differential equation is a particular case of the above equation, with ν =1, whereas the

3

Gompertz curve can be recovered in the limit  ν  provided that:

→

0+

5

References •

α=O

�1� ν 

In fact, for small ν it is

′

Y  (t) = Y r

Y   K 

� � ��

1 − exp ν  ln

≈

ν 

rY  ln

� � Y  K 

•

  Pella, J. S.; Tomlinson, P. K. (1969). “A Generalised Stock-Production Model”.   Bull. Inter-Am. Trop. Tuna Comm.  13 : 421–496.

•

  Lei, Y. C.; Zhang, S. Y. (2004). “Features and Partial Derivatives of Bertalanffy–Richards Growth Model in Forestry”.  Nonlinear Analysis: Modelling and Control . 9  (1): 65–73.

The RDE suits to model many growth phenomena, including the growth of tumours. Concerning its applications in oncology, its main biological features are similar to those of Logistic curve model.

6 2

Gradient of generalized logistic function

When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point t (see  [1] ). For the case where C  = 1 ,

∂Y  = 1 − (1 + Qe B (t M ) ) 1/ν  ∂A ∂Y   = (1 + Qe B (t M ) ) 1/ν  ∂K  (K  − A)(t − M )Qe B (t M ) ∂Y  = 1 ∂B ν (1 + Qe B (t M ) ) +1 −

−

−

−

−

−

−

−

−

−

ν

−B

(K  − A) ln(1 + Qe ∂Y  = ∂ν  ν 2 (1 + Qe B (t

(t−M )

))

−M 

−

)

1 ν

(K  − A)e B (t M ) ∂Y  =− 1 ∂Q ν (1 + Qe B (t M ) ) +1 −

−

−

−

(K  − A)QBe ∂Y  =− ∂M  ν (1 + Qe B (t −

3

4

−B

ν

(t−M ) ) ) 1 +1

−M 

ν

See also •

  Logistic function

•

 Gompertz curve

•

  Ludwig von Bertalanffy

Footnotes

[1] Fekedulegn, Desta; Mairitin P. Mac Siurtain; Jim J. Colbert (1999). “Parameter Estimation of Nonlinear Growth Models in Forestry” (PDF).  Silva Fennica. 33   (4): 327– 336. Retrieved 2011-05-31.

Richards, F. J. (1959). “A Flexible Growth Function for Empirical Use”. Journal of  10 (2): 290–300. Experimental Botany. doi:10.1093/jxb/10.2.290.

External links •

 YAN Kun(2011). Research on adaptive connection equation in discontinuous area of data curve( General tendency equation of natural saturation process curve and creep process curve), DOI:10.3969/j.issn. 1004-2903.2011.01.018.

4

7

7

TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

Text and image sources, contributors, and licenses

7.1 •

7.2 •

•

•

•

•

•

•

7.3 •

Text   Generalised logistic function   Source:    https://en.wikipedia.org/wiki/Generalised_logistic_function?oldid=770138149   Contributors:  JohnOwens, Michael Hardy, Wapcaplet, Charles Matthews, David Gerard, BenFrantzDale, Bender235, HenkvD, Rjwilmsi, TeaDrinker, BradBeattie, Mikeblas, Can't sleep, clown will eat me, Fidlej, Baccyak4H, LokiClock, Masterken, Bmcm, Vvnataraj, DFRussia, Phipperdee, Genie05, Bbanerje, Fgnievinski, New Image Uploader 929, Rhaertel80, Yobot, Julia W, Some standardized rigour, Debiatan, RjwilmsiBot, Tdc204, Reverend T. R. Malthus, Khazar2, Monkbot, Mikelonergan7, Armavica and Anonymous: 18

Images  File:GeneralizedLogisticA.svg   Source:    https://upload.wikimedia.org/wikipedia/en/9/98/GeneralizedLogisticA.svg   License:    CC-BYSA-3.0 Contributors:  ?  Original artist:  ?  File:GeneralizedLogisticB.svg Source:  https://upload.wikimedia.org/wikipedia/en/d/d0/GeneralizedLogisticB.svg License:   CC-BY-SA3.0  Contributors:  ?  Original artist:  ?  File:GeneralizedLogisticC.svg   Source:    https://upload.wikimedia.org/wikipedia/en/8/8b/GeneralizedLogisticC.svg   License:    CC-BYSA-3.0 Contributors:  ?  Original artist:  ?  File:GeneralizedLogisticK.svg Source:  https://upload.wikimedia.org/wikipedia/en/c/cc/GeneralizedLogisticK.svg License:   CC-BY-SA3.0  Contributors:  ?  Original artist:  ?  File:GeneralizedLogisticNu.svg  Source:  https://upload.wikimedia.org/wikipedia/en/d/d1/GeneralizedLogisticNu.svg License:  CC-BYSA-3.0 Contributors:  ?  Original artist:  ?  File:GeneralizedLogisticQ.svg Source:  https://upload.wikimedia.org/wikipedia/en/c/cf/GeneralizedLogisticQ.svg License:   CC-BY-SA3.0  Contributors:  ?  Original artist:  ? File:Generalized_logistic_function_A0_K1_B1.5_Q0.5_ν0.5_M0.5.png   Source:    https://upload.wikimedia.org/wikipedia/commons/ 5/54/Generalized_logistic_function_A0_K1_B1.5_Q0.5_%CE%BD0.5_M0.5.png License:   Public domain Contributors:  Transferred from en.wikipedia to Commons by Ronhjones.  Original artist:   Debiatan at en.wikipedia

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Smoothstep polynomial in the general smoothstep is 2 N +1. With N  = 1, the slopes or first derivatives of the smoothstep are equal to zero at the left and right edge (x=0 and x=1) where the curve is appended to the constant or saturated levels. With higher integer  N , the second and higher derivatives are zero at the edges making the polynomial functions as flat as possible making the splice to the limit values of 0 or 1 more seamless.

1 smoothstep(x) smootherstep(x)

0.9 0.8 0.7 0.6 0.5 0.4 0.3

In   MSDN and   OpenGL   libraries,   smoothstep   implements the S1 (x)  , the cubic  Hermite interpolation after clamping:

0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 ( )= 3 1

A plot of the smoothstep(x) and smootherstep(x) functions, using 0 as the left edge and 1 as the right edge.

smoothstep(x) =  S 1 x

Smoothstep is a family of sigmoid-like interpolation and clamping  functions commonly used in  computer graphics[1][2] and video game engines.[3]

SN  x

x

N 

n=0

N +n

2N +1

n

N −n

S0 x

x

0≤x≤1 1≤x

float smoothstep(float edge0, float edge1, float x) { // Scale, bias and saturate x to 0..1 range x = clamp((x edge0)/(edge1 - edge0), 0.0, 1.0); // Evaluate polynomial return x*x*(3 - 2*x); } float clamp(float x, float lowerlimit, float upperlimit) { if (x < lowerlimit) x = lowerlimit; if (x > upperlimit) x = upperlimit; return x; }

1

Variations

Ken Perlin suggests[5] an improved version of the smoothstep function which has zero 1st and 2nd order derivatives at x =0 and  x =1:

ifx ≤ 0 n

−x)

S0 (x)  is identical to the clamping function:

0 ( )= 1

x − 2x

A C/C++ example implementation provided by AMD [4] follows.

With no salient loss of generality, the left edge may be set to 0 and the right edge to 1. Then the general form for smoothstep is

N +1

3

Again assuming that the left edge is 0, the right edge is 1, with the transition between edges taking place where 0 ≤  x  ≤ 1.

The function depends on three parameters, the input  x , the “left edge” and the “right edge”, with the left edge being assumed smaller than the right edge. The function receives a real number  x  as an argument and returns 0 if  x  is less than or equal to the left edge, 1 if x is greater than or equal to the right edge, and smoothly interpolates, using a Hermite polynomial, between 0 and 1 otherwise. The slope of the   smoothstep  function is zero at both edges. This is convenient for creating a sequence of transitions using  smoothstep  to interpolate each segment as an alternative to using more sophisticated or expensive interpolation techniques.

0 ∑ � �� �( ( )= 1

x≤0 2

if0 ≤ x ≤ 1 if1 ≤ x smootherstep(x) =  S 2 (x) =

0 61

x≤0 5

4

3

x − 15x + 10x

C/C++ reference implementation:

ifx ≤ 0

float smootherstep(float edge0, float edge1, float x) { // Scale, and clamp x to 0..1 range x = clamp((x edge0)/(edge1 - edge0), 0.0, 1.0); // Evaluate polynomial return x*x*x*(x*(x*6 - 15) + 10); } float clamp(float x, float lowerlimit, float upperlimit) { if (x < lowerlimit) x

if0 ≤ x ≤ 1 if1 ≤ x

The characteristic “S"-shaped sigmoid curve is obtained with SN (x)  only for integers N   ≥ 1. The order of the 1

0≤x≤1 1≤x

2

2

= lowerlimit; if (x > upperlimit) x = upperlimit; return x; }

2

f (0) = 0

⇒

0 + 0 + 0 + 0 + 0 + a 0  = 0

f (1) = 1

⇒

a5  +  a 4  +  a 3  +  a 2  +  a 1  +  a 0  = 1

Applying the desired values for the first derivative of the function at both endpoints we get:

Origin

2.1

ORIGIN 

3rd order equation

We start with a generic third order  polynomial function and its first derivative:

f ′ (x) = 3 a3 x2 + 2 a2 x  + a1 Applying the desired values for the function at both endpoints we get:

⇒

f (1) = 1

⇒

⇒

f ′ (1) = 0

⇒

0 + 0 + 0 + 0 + a 1  = 0 5a5  + 4a4  + 3a3  + 2a2  +  a 1  = 0

Applying the desired values for the second derivative of the function at both endpoints we get:

f (x) = a3 x3 +  a 2 x2 +  a 1 x  +  a 0

f (0) = 0

f ′ (0) = 0

f ′′ (0) = 0

⇒

f ′′ (1) = 0

⇒

0+ 0 + 0 + 2a2  = 0 20a5  + 12a4  + 6a3  + 2a2  = 0

Solving the system of 6 unknowns formed by the last 6 equations we obtain the values of the polynomial coefficients:

0 + 0 + 0 +  a 0  = 0 a3  +  a 2  +  a 1  +  a 0  = 1

Applying the desired values for the first derivative of the function at both endpoints we get:

a0  = 0,

a1  = 0,

a2  = 0,

a3  = 10,

a4  = −15,

a5  = 6

Introducing these coefficients back into the first equation gives the fifth order smootherstep function: f ′ (0) = 0 ′

f  (1) = 0

0 + 0 +  a 1  = 0 3a3  + 2a2  +  a 1  = 0

⇒ ⇒

Solving the system of 4 unknowns formed by the last 4 equations we obtain the values of the polynomial coefficients:

a0  = 0,

a1  = 0,

a2  = 3,

a3  = −2

f (x) = 6x5 − 15x4 + 10x3

2.3

7th order equation

Also called “smootheststep”, the 7th order equation was derived by Kyle McDonald  and first posted to Twitter[6] with a derivation on GitHub:[7]

Introducing these coefficients back into the first equation gives the third order smoothstep function: f (x) = −20x7 + 70x6 − 84x5 + 35x4 f (x) = −2x3 + 3x2

2.4 2.2

5th order equation

We start with a generic fifth order  polynomial function, its first derivative and its second derivative:

Generalization of higher-order equations

Smoothstep polynomials are generalized, with 0≤ x ≤1 as:

N 

f (x) = ′

5

a5 x + 4

4

3

3

2

2

′′

3

SN  x

a4 x + a3 x +  a 2 x +  a 1 x  +  a 0

f  (x) = 5a5 x + 4a4 x + 3 a3 x + 2a2 x  + a1 2

f  (x) = 20a5 x + 12a4 x + 6a3 x  + 2a2

�  + ��2  + 1� � ( )= ( �  + ��2  + 1� � = ( 1) � 1��2  + 1� � =  x

−

n

N 

n

n=0

n=0

−N  −

n

n

N  N  − n

n

N  N  − n

n

n=0

N 

N 

Applying the desired values for the function at both endpoints we get:

N 

N +1

N  N  − n

n

−x)

xN +n+1

xN +n+1

N  ∈ Z ≥ 0

3

where N  determines the order of the resulting polynomial function, which is 2N +1. The first seven Smoothstep polynomials, with 0≤ x ≤1, are expressed as:

[6] kcimc (25 March 2015). “smootheststep(t)=−20*t^7+70*t^6-84*t^5+35*t^4 // when smootherstep’s second derivative isn't enough” (Tweet) – via Twitter. [7] Kyle McDonald (27 March 2015).   “Derivation of 7thorder smoothstep function with zeros in third derivative.”. Github.com. Retrieved 20 December 2015.

S0 (x) =  x S1 (x) = −2x3 + 3x2 S2 (x) = 6x5 − 15x4 + 10x3

[8]   http://stackoverflow.com/questions/41195063/ general-smoothstep-equation/

S3 (x) = −20x7 + 70x6 − 84x5 + 35x4 S4 (x) = 70x9 − 315x8 + 540x7 − 420x6 + 126x5

462x6 S5 (x) = −252x11 + 1386x10 − 3080x9 + 3465x8 − 1980 4x7 +External 13

12

S6 (x) = 924x

− 6006x

+ 16380x

11

− 24024x

10

+ 20020x

It can be shown that the Smoothstep polynomials S N (x) that transition from 0 to 1 when  x  transitions from 0 to 1 can be simply mapped to  odd-symmetry polynomials,

9

− 9009x

1

RN  (x) =

−1

1−u

0

x

� � ∫  �1 �

2 N 

2 N 

du

−u

du

0

where

SN (x) = 12  R N (2x − 1) +

1 2

and RN (−x) = − RN (x)  . The argument of R N (x) is −1≤ x ≤1 and is appended to the constant −1 on the left and +1 at the right. An implementation of SN (x)  in Javascript:[8] function generalSmoothStep(N, x) //Generalized smoothstep { x = clamp(x, 0, 1); //x must be equal to or between 0 and 1 var result = 0; for (var n=0; n<=N; n++) { result += (pascalTriangle(-N-1, n) * pascalTriangle(2*N+1, N-n) * Math.pow(x, N+n+1)); } return result; } function pascalTriangle(a, b) { //Returns binomial coefficient without explicit use of factorials which can't be used with negative integers var result = 1; for(var i=0; i upperlimit) x = upperlimit; return x; }

3

References

[1] Smoothstep at Microsoft Developer Network [2] GLSL Language Specification, Version 1.40 [3] Unity game engine SmoothStep documentation [4]  ATI R3x0 Pixel Shaders [5]   Texturing and Modeling, Third Edition: A Procedural Approach

links + 1716x7

•  Using smoothstep (in the RenderMan Shading Lan-

guage) by Prof. Malcolm Kesson. •  Interpolation tricks by Jari Komppa • Swift

�∫  �

8

Interpolation Playground   demonstrates smoothStep(), smootherStep()  and  smoothestStep() in a Swift playground by Simon Gladman

4

5

5

TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

Text and image sources, contributors, and licenses

5.1 •

5.2 •

5.3 •

Text   Smoothstep  Source:   https://en.wikipedia.org/wiki/Smoothstep?oldid=781117381 Contributors:  AxelBoldt, Axlrosen, Brighterorange, Bgwhite, Malcolma, MoritzMoeller, Magioladitis, Addbot, Yobot, Tcpp, Otaviogood, Erik9bot, FrescoBot, Magwo001, Cnwilliams, DanielBrauer, BG19bot, Smoothstep graph, MathieuRouvinez, Cycloverid, Narky Blert, Trung0246 and Anonymous: 25

Images File:Smoothstep_and_Smootherstep.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/5/57/Smoothstep_and_ Smootherstep.svg License:  CC0  Contributors:   Own work  Original artist:  Smoothstep graph

Content license Creative Commons Attribution-Share Alike 3.0

Algebraic function This article is about algebraic functions in   calculus, An  algebraic function in m  variables  is similarly demathematical analysis, and abstract algebra. For func- fined as a function y  which solves a polynomial equation tions in elementary algebra, see function (mathematics). in m  + 1 variables: In mathematics, an  algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.

 p(y, x1 , x2 , . . . , xm ) = 0.

It is normally assumed that p  should be an   irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in m  variables over the field  K  is an element of the algebraic closure of the field of rational functions  K ( x 1 ,..., xm).

Examples of such functions are:

• • •

f (x) = 1/x f (x) = f (x) =

√ x 1

√ 

3 1+ √ x x3/7 − 7x1/3

Algebraic functions in one variable

Some algebraic functions, however, cannot be expressed by such finite expressions (this is Abel–Ruffini theorem). 1.1 Introduction and overview This is the case, for example, of the  Bring radical, which is the function implicitly defined by The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic f (x)5 + f (x) +  x  = 0 functions as functions which can be formed by the usual algebraic operations:   addition,  multiplication, division, In more precise terms, an algebraic function of degree  n and taking an  n th root. This is something of an oversimin one variable  x  is a function  y = f (x) that satisfies a plification; because of the  fundamental theorem of Gapolynomial equation lois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function  y = p(x)  is an algebraic function, since it is simply the solution  y  to the where the coefficients  ai ( x ) are polynomial functions of equation  x , with coefficients belonging to a set  S . Quite often,  S  = Q , and one then talks about “function algebraic over Q ", and the evaluation at a given rational value of such an y − p(x) = 0. algebraic function gives an algebraic number. p(x) A function which is not algebraic is called a More generally, any rational function  y = q(x)  is algetranscendental function, as it is for example the braic, being the solution to case of exp(x), tan(x), ln(x), Γ(x)  . A composition of transcendental functions can √ give an algebraic function: f (x) = cos(arcsin(x)) = 1 − x2 . q (x)y −  p(x) = 0. an (x)y n + an−1 (x)y n−1 +

· ·· + a0(x) = 0

As an equation of degree n has n   roots, a polynomial equation does not implicitly define a single function, but n Moreover, the  nth root of any polynomial y  = functions, sometimes also called branches. Consider for an algebraic function, solving the equation example the equation of the unit circle: y2 +  x 2 = 1. This determines  y , except only up to an √ overall sign; ac- n cordingly, it has two branches: y  = ± 1 − x2. y −  p(x) = 0. 1

√  p(x) is n

2

1

ALGEBRAIC FUNCTIONS IN ONE VARIABLE 

Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that  y is a solution 3 y − xy + 1 = 0. to Using the cubic formula, we get an (x)y n +

·· · + a0(x) = 0 ,

2x

√ −108 + 12√ 81 − 12x 3

for each value of  x , then  x  is also a solution of this equa- y  = − 3 √  − 3 + 6 − 108 + 12 81 12x tion for each value of  y . Indeed, interchanging the roles of x  and  y  and gathering terms, 3 For  x  ≤ √  3 ,  the square root is real and the cubic root 4 is thus well defined, providing the unique real root. On 3 the other hand, for  x > √  3 ,  the square root is not real, 4 bm (y )xm + bm−1 (y )xm−1 + · ·· + b0 (y ) = 0. and one has to choose, for the square root, either nonWriting x  as a function of y  gives the inverse function, real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are also an algebraic function. done in the two terms of the formula, the three choices However, not every function has an inverse. For example, for the cubic root provide the three branches shown, in y  =  x 2 fails the horizontal line test: it fails to be one-to- the accompanying image. √  one. The inverse is the algebraic “function”  x  =  ± y  . Another way to understand this, is that the set of branches It may be proven that there is no way to express this funcof the polynomial equation defining our algebraic func- tion in terms of  nth  roots using real numbers only, even though the resulting function is real-valued on the domain tion is the graph of an algebraic curve. of the graph shown.

√ 

On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complex analysis to discuss algebraic functions. In parFrom an algebraic perspective, complex numbers enter ticular, the argument principle can be used to show that quite naturally into the study of algebraic functions. First any algebraic function is in fact an  analytic function, at of all, by the fundamental theorem of algebra, the com- least in the multiple-valued sense. plex numbers are an algebraically closed field. Hence any polynomial relation  p (y,  x ) = 0 is guaranteed to have at Formally, let  p( x , y) be a complex polynomial in the comleast one solution (and in general a number of solutions plex variables  x  and  y . Suppose that  x 0  ∈  C  is such that not exceeding the degree of  p  in  x ) for  y  at each point  x , the polynomial  p( x 0 ,y) of y has n  distinct zeros. We shall show that the algebraic function is analytic in a neighborprovided we allow  y  to assume complex as well as real values. Thus, problems to do with the domain of an alge- hood of  x 0 . Choose a system of  n  non-overlapping discs Δi  containing each of these zeros. Then by the argument braic function can safely be minimized. principle 1.2

The role of complex numbers

1 2πi

 

∂ ∆i

 py (x0 , y ) dy  = 1.  p(x0 , y)

By continuity, this also holds for all  x   in a neighborhood of  x 0 . In particular,  p ( x ,y) has only one root in Δi , given by the residue theorem: 1 f i (x) = 2πi A graph of three branches of the algebraic function  y , where  y − xy  + 1 = 0, over the domain 3/2 2/3 < x  < 50.

3

Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express the function in terms of addition, multiplication, division and taking  nth  roots without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation

 

∂ ∆i

 py (x, y ) y dy  p(x, y )

which is an analytic function. 1.3

Monodromy

Note that the foregoing proof of analyticity derived an expression for a system of  n  different  function elements  fi ( x ), provided that x   is not a  critical point  of  p ( x , y). A  critical point  is a point where the number of distinct

3

.

3 zeros is smaller than the degree of  p, and this occurs only where the highest degree term of  p  vanishes, and where the  discriminant vanishes. Hence there are only finitely many such points  c 1 , ...,  cm .

•  Rational function •  Special functions •  Transcendental function

A close analysis of the properties of the function elements  fi  near the critical points can be used to show that the monodromy cover is ramified over the critical points (and 4 References possibly the point at infinity). Thus the holomorphic extension of the fi  has at worst algebraic poles and ordinary •  Ahlfors, Lars (1979).  Complex Analysis . McGraw algebraic branchings over the critical points. Hill. Note that, away from the critical points, we have •   van der Waerden, B.L.  (1931). Modern Algebra, Volume II . Springer.  p(x, y ) =  a n (x)(y

− f 1(x))(y − f 2(x)) · ·· (y − f  (x)) n

since the  f i  are by definition the distinct zeros of  p . The 5 External links monodromy group acts by permuting the factors, and •  Definition of “Algebraic function” in the Encyclothus forms the monodromy representation of the Galois pedia of Math group of p. (The monodromy action on the universal covering space is related but different notion in the theory of • Weisstein, Eric W. “Algebraic Function”. Riemann surfaces.) MathWorld .

2

History

The ideas surrounding algebraic functions go back at least as far as René Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge  in which he writes: let a quantity denoting the ordinate, be an algebraic function of theabscissax, bythecommon methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms.

3

See also

•  Algebraic expression •  Analytic function •  Complex function •  Elementary function •   Function (mathematics) •  Generalized function •  List of special functions and eponyms • List of types of functions •   Polynomial

•  Algebraic Function at PlanetMath.org. •  Definition of “Algebraic function” in David J. Darling's Internet Encyclopedia of Science