Erf

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

erf(z) is an entire analytical function of z which is defined in the whole complex z-plane.

Symmetries and periodicities

Parity

erf(z) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function erf(z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function erf(z) does not have branch points.

Branch cuts

The function erf(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Asymptotic series expansions

Residue representations

Other series representations

Limit representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Continued fraction representations

Involving the function

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and elementary functions

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Laplace transforms

Mellin transforms

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1F1

Involving HypergeometricPFQ

Involving hypergeometric HypergeometricU

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving Erfi

Generalized cases for the direct function itself

Generalized cases involving Exp

Generalized cases involving Erfi

Through other functions

Representations through equivalent functions

With inverse function

With related functions

Zeros

Theorems

Limit theorem of de Moivre-Laplace

In a large number of Bernoulli trials, the probability that a random variable X with a standard binomial distribution P(k)=(({{n}, {k}}))(((p^k)(1-p))^(n-k)) takes on a value a≤x≤b is given by

(lim_(n->∞))P(a≤((X-n p)/(Sqrt[n p(1-p)]))≤b)==(1/2)(erf((b/(Sqrt[2])))-erf((a/(Sqrt[2])))).

The smoothing function occuring in hyperasymptotic expansions across Stokes lines

The smoothing function s(ξ) occuring in hyperasymptotic expansions across Stokes lines is of the universal  form  s(ξ)==(1/2)(1+erf(ξ)).

History

– A. de Moivre (1718,1733)
– P.-S. Laplace (1774)
– A.de Moivre (1788); 
– P.-S. Laplace (1812) derived an asymptotic expansion

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