Erfc

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

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

Branch points

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

Branch cuts

The function erfc(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

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

Involving direct function and Gamma-, Beta-, Erf-type functions

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Hankel transforms

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1F1

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

Theorems

Solution of the one-dimensional heat equation

The function w(x,t)==erfc((x/(2 (Sqrt[t])))) fulfills the one-dimensional heat equation ((∂w(x,t))/(∂t))==(((∂^2)w(x,t))/(∂(x^2))).

The iterated integral of the function erfc(z)

The iterated integral of the function erfc(z) can be expressed polynomially in erfc(z), exp(z), and z.

The solution to the initial value problem for the time-dependent Schrödinger equation

The functions  (Subscript[ψ, k])(x,t)=(e^(i (k x-(k^2) t/2))) erfc((e^(-((i π)/4))) (x-k t)/(Sqrt[2 t])) are a solution to the initial value problem (Subscript[ψ, k])(x,0)=θ(-x)(e^(i k x)) for the time–dependent Schrödinger equation i ((∂ψ(x,t))/(∂t))==-(1/2) (((∂^2)ψ(x,t))/(∂(x^2))).

History

– C. Kramp (1799)

Copyright