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playstyle frac 1 2 frac 1 pi arctan x a 1 2 1 π arctan a 2 x b 1 x b 1 x b displaystyle frac 1 2 frac 1 pi arctan left frac a 2x b 1 x b 1 x b right 7 hyperbolic 1 2 1 2 tanh a x displaystyle frac 1 2 frac 1 2 tanh a x 1 2 1 2 tanh a ln x displaystyle frac 1 2 frac 1 2 tanh a ln x log logistic 8 1 2 1 2 tanh a 2 x b 1 x b 1 x b displaystyle frac 1 2 frac 1 2 tanh left frac a 2x b 1 x b 1 x b right 8 inverse hyperbolic 2 π arcsin 1 1 e a x c e c 2 1 displaystyle frac 2 pi arcsin left frac 1 1 e a x c right e c sqrt 2 1 2 π arcsin x a 1 x a displaystyle frac 2 pi arcsin left frac x a 1 x a right 2 π arcsin 1 1 e a cot π x displaystyle frac 2 pi arcsin left frac 1 1 e a cot pi x right 9 special 1 2 1 2 erf a 2 x displaystyle frac 1 2 frac 1 2 operatorname erf left frac a sqrt 2 x right normal 9 10 1 2 1 2 erf a 2 ln x displaystyle frac 1 2 frac 1 2 operatorname erf left frac a sqrt 2 ln x right log normal 11 1 2 1 2 erf a 2 x b 1 x b 1 x b displaystyle frac 1 2 frac 1 2 operatorname erf left frac a 2x b 1 x b 1 x b right 10 stochastic e a b 2 2 x b w i y 0 1 a 0 x e a b 2 2 s b w s d s 1 displaystyle e left a frac b 2 2 right x b w_ i cdot left y_ 0 1 a int _ 0 x e left a frac b 2 2 right s b w_ s ds right 1 11 chaotic d x d t x y z d y d t a y a z d z d t z b y b 1 x y displaystyle frac dx dt x yz quad frac dy dt ay az quad frac dz dt z by b 1 xy grebenc 12 table 2 taxonomy matrix with examples of sigmoid functions of the 2 nd kind on the unbounded interval no type of sigmoid function unbounded interval x a 1 b 0 a b 1 displaystyle x in infty infty a 1 b 0 a b 1 explanation 1 all functions from table 1 s a x a ai b x displaystyle s a x a cdot operatorname ai b x adding a ai b x displaystyle a cdot operatorname ai bx to the functions s a x displaystyle s ax of the 3 rd column of the table 1 where ai displaystyle operatorname ai is the airy ai function 2 special a ai a x d x 1 3 displaystyle a int operatorname ai a x dx tfrac 1 3 3 special a s a x b 8 1 2 c a x 2 b 8 1 2 s a x 2 displaystyle a s a x frac b 8 left 1 2 c a x right 2 frac b 8 left 1 2 s a x right 2 c displaystyle c s displaystyle s are fresnel integrals 4 special 1 sin b e a x b e a x displaystyle 1 frac sin b e a x b e a x 5 special 1 j 0 b e a x displaystyle 1 j_ 0 left b e a x right j 0 displaystyle j_ 0 is a bessel j function table 3 taxonomy matrix with examples of sigmoid functions of the 3 rd kind no type unbounded interval x a b 1 b 0 displaystyle x in infty infty a b 1 b neq 0 explanation 1 special a s a x b 1 2 si a x π displaystyle a s a x b left tfrac 1 2 frac operatorname si ax pi right adding b 1 2 si a x π displaystyle b left frac 1 2 frac operatorname si ax pi right to the functions a s a x displaystyle a s ax of the 3 rd column of the table 1 where si displaystyle operatorname si is the sine integral function 2 special a s a x b 1 2 1 2 e ci a x displaystyle a s a x b left tfrac 1 2 tfrac 1 2 e operatorname ci ax right adding b 1 2 e ci a x 2 displaystyle b left frac 1 2 frac e operatorname ci ax 2 right to the functions a s a x displaystyle a s ax of the 3 rd column of table 1 where ci displaystyle operatorname ci is the cosine integral function 3 special a s a x b 1 2 s a x displaystyle a s a x b left tfrac 1 2 s a x right s a x displaystyle s ax is the fresnel s integral 4 special a s a x b 1 2 c a x displaystyle a s a x b left tfrac 1 2 c a x right c a x displaystyle c ax is the fresnel c integral construction methods edit the same theory presents a list of 30 methods for constructing sigmoid functions 1 these include algebraic transformations integration and convolution methods constructions from bell shaped functions solutions of ordinary and partial differential equations recursive schemes stochastic differential equations feedback systems and chaotic systems m0 construction method for sigmoid functions not evident or intuitive m1 inverse of singularity functions m2 sigmoid functions of embedded positive functions m3 rising a sigmoid function to the power m4 exponentiating a sigmoid function m5 symmetric sigmoid functions derived from asymmetric ones m6 sigmoid functions of the reciprocal independent variable m7 embedding a sigmoid function into other function m8 sum of sigmoid functions m9 multiplication of sigmoid functions m10 integral of the product of an increasing and a decreasing function m11 derivation from lambda bell shaped functions m12 integration of lambda bell shaped function m13 integration of the sum of lambda bell shaped functions m14 integration of the product of two lambda bell shaped functions m15 integration of the difference of two shifted sigmoid functions m16 integration of the product of two shifted sigmoid functions m17 convolution of sigmoid functions m18 integration of the product of lambda and sigmoid function m19 solutions of ordinary differential equations m20 solutions of partial differential equation pde m21 solutions of functional differential equation fde m22 sum of a sigmoid function and some derivatives m23 combination of sigmoid functions its derivative and integral m24 filtering sigmoid functions m25 special cases of gauss hypergeometric functions m26 feedback closed loop systems m27 recursive functions m28 recursive time delayed feed forward loops m29 solutions of stochastic differential equation m30 chaotic sigmoid functions consult reference 1 for more details definition edit a sigmoid function is a bounded differentiable real function that is defined for all real input values and has a positive derivative at each point 13 14 properties edit in general a sigmoid function is monotonic and has a first derivative which is bell shaped conversely the integral of any continuous non negative bell shaped function with one local maximum and no local minimum unless degenerate will be sigmoidal thus the cumulative distribution functions for many common probability distributions are sigmoidal one such example is the error function which is related to the cumulative distribution function of a normal distribution another is the arctan function which is related to the cumulative distribution function of a cauchy distribution a sigmoid function is constrained by a pair of horizontal asymptotes as x displaystyle x rightarrow pm infty a sigmoid function is convex for values less than a particular point and it is concave for values greater than that point in many of the examples here that point is 0 examples edit some sigmoid functions compared in the drawing all functions are normalized in such a way that their slope at the origin is 1 logistic function f x 1 1 e x displaystyle f x frac 1 1 e x hyperbolic tangent shifted and scaled version of the logistic function above f x tanh x e x e x e x e x displaystyle f x tanh x frac e x e x e x e x arctangent function f x arctan x displaystyle f x arctan x gudermannian function f x gd x 0 x d t cosh t 2 arctan tanh x 2 displaystyle f x operatorname gd x int _ 0 x frac dt cosh t 2 arctan left tanh left frac x 2 right right error function f x erf x 2 π 0 x e t 2 d t displaystyle f x operatorname erf x frac 2 sqrt pi int _ 0 x e t 2 dt generalised logistic function f x 1 e x α α 0 displaystyle f x left 1 e x right alpha quad alpha 0 smoothstep function f x 0 1 1 u 2 n d u 1 0 x 1 u 2 n d u x 1 sgn x x 1 n z 1 displaystyle f x begin cases displaystyle left int _ 0 1 left 1 u 2 right n du right 1 int _ 0 x left 1 u 2 right n du x leq 1 operatorname sgn x x geq 1 end cases quad n in mathbb z geq 1 some algebraic functions for example f x x 1 x 2 displaystyle f x frac x sqrt 1 x 2 and in a more general form 15 f x x 1 x k 1 k displaystyle f x frac x left 1 x k right 1 k up to shifts and scaling many sigmoids are special cases of f x φ φ x β α displaystyle f x varphi varphi x beta alpha where φ x λ 1 λ x 1 λ λ 0 e x λ 0 displaystyle varphi x lambda begin cases 1 lambda x 1 lambda lambda neq 0 e x lambda 0 end cases is the inverse of the negative box cox transformation and α 1 displaystyle alpha 1 and β 1 displaystyle beta 1 are shape parameters 16 smooth transition function 17 normalized to 1 1 f x 2 1 e 2 m x 1 x 2 1 x 1 sgn x x 1 tanh m x 1 x 2 x 1 sgn x x 1 displaystyle begin aligned f x begin cases displaystyle frac 2 1 e 2m frac x 1 x 2 1 x 1 operatorname sgn x x geq 1 end cases begin cases displaystyle tanh left m frac x 1 x 2 right x 1 operatorname sgn x x geq 1 end cases end aligned using the hyperbolic tangent mentioned above here m displaystyle m is a free parameter encoding the slope at x 0 displaystyle x 0 which must be greater than or equal to 3 displaystyle sqrt 3 because any smaller value will result in a function with multiple inflection points which is therefore not a true sigmoid this function is unusual because it actually attains the limiting values of 1 and 1 within a finite range meaning that its value is constant at 1 for all x 1 displaystyle x leq 1 and at 1 for all x 1 displaystyle x geq 1 nonetheless it is smooth infinitely differentiable c displaystyle c infty everywhere including at x 1 displaystyle x pm 1 applications edit inverted logistic s curve to model the relation between wheat yield and soil salinity many natural processes such as those of complex systems learning curves exhibit a progression from small beginnings that accelerates and approaches a climax over time 18 when a specific mathematical model is lacking a sigmoid function is often used 19 the van genuchten gupta model is based on an inverted s curve and applied to the response of crop yield to soil salinity examples of the application of the logistic s curve to the response of crop yield wheat to both the soil salinity and depth to water table in the soil are shown in modeling crop response in agriculture in artificial neural networks non smooth functions are sometimes used for efficiency these are known as hard sigmoids in audio signal processing sigmoid functions are used as waveshaper transfer functions to emulate the sound of analog circuitry clipping 20 in digital signal processing in general sigmoid functions due to their higher order of continuity have much faster asymptotic rolloff in the frequency domain than a heavyside step function and therefore are useful to smooth discontinuities before sampling to reduce aliasing this is for example used to generate square waves in many kinds of digital synthesizer in biochemistry and pharmacology the hill and hill langmuir equations are sigmoid functions in driving changing lanes is accomplished smoothly by following a sigmoid trajectory in computer graphics and real time rendering sigmoid functions are used to blend colors or geometry between two values producing smooth transitions without visible seams or discontinuities titration curves between strong acids and strong bases have a sigmoid shape due to the logarithmic nature of the ph scale the logistic function can be calculated efficiently by utilizing type iii unums 21 a hierarchy of sigmoid growth models with increasing complexity number of parameters was built 22 with the primary goal to re analyze kinetic data the so called n t curves from heterogeneous nucleation experiments 23 in electrochemistry the hierarchy includes at present three models with 1 2 and 3 parameters if not counting the maximal number of nuclei n max respectively a tanh 2 based model called α 21 24 originally devised to describe diffusion limited crystal growth not aggregation in 2d the johnson mehl avrami kolmogorov jmak model 25 and the richards model 26 it was shown that for the concrete purpose even the simplest model works and thus it was implied that the experiments revisited are an example of two step nucleation with the first step being the growth of the metastable phase in which the nuclei of the stable phase form 22 see also edit wikimedia commons has media related to sigmoid functions step function linear combination of indicator functions of real intervals sign function function returning minus 1 zero or plus 1 heaviside step function indicator function of positive numbers logistic regression statistical model for a binary dependent variable logit function in statistics softplus function smoothed ramp function pages displaying short descriptions of redirect targets softmax function smooth approximation of one hot arg max swish function mathematical activation function in data analysis weibull distribution continuous probability distribution fermi dirac statistics statistical description for the behavior of fermions references edit 1 2 3 grebenc andrej august 2025 foundations for a general theory of sigmoid functions modelling with 30 methods intelligent systems conference cham springer nature switzerland pp 481 511 doi 10 1007 978 3 031 99958 1 elliot d l 1993 a better activation function for artificial neural networks technical report college park md institute for systems research university of maryland tr 93 8 richards f j 1959 a flexible growth function for empirical use journal of experimental botany 10 2 290 300 doi 10 1093 jxb 10 2 290 verhulst p f 1838 notice sur la loi que la population suit dans son accroissement correspondance mathématique et physique 10 113 121 laplace p s 1774 mémoire sur la probabilité des causes par les évènements mémoires de l académie royale des sciences présentés par divers savans 6 621 656 rayleigh l 1880 on the resultant of a large number of vibrations of the same pitch and of arbitrary phase philosophical magazine 5th series 10 73 78 cauchy a l 1853 memoire sur les resultats moyens d un tres grand nombre des observations comptes rendus hebdomadaires des séances de l académie des sciences 37 381 385 fisk p r 1961 the graduation of income distributions econometrica 29 2 171 185 doi 10 2307 1909287 gauss c f 1809 theoria motvs corporvm coelestivm in sectionibvs conicis solem ambientivm hamburg perthes et besser glaisher james whitbread lee 1871 on a class of definite integrals london edinburgh and dublin philosophical magazine and 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