15.6 Error Function

The Error function and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7

Function: erf (z) ¶

The Error Function erf(z) (A&S 7.1.1)

See also flag erfflag.

‘Category: Special functions’

Function: erfc (z) ¶

The Complementary Error Function erfc(z) (A&S 7.1.2)

erfc(z) = 1-erf(z)

‘Category: Special functions’

Function: erfi (z) ¶

The Imaginary Error Function.

erfi(z) = -%i*erf(%i*z)

‘Category: Special functions’

Function: erf_generalized (z1,z2) ¶

Generalized Error function Erf(z1,z2)

‘Category: Special functions’

Function: fresnel_c (z) ¶

The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1)

The simplification fresnel_c(-x) = -fresnel_c(x) is applied when flag trigsign is true.

The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when flag %iargs is true.

See flags erf_representation and hypergeometric_representation.

‘Category: Special functions’

Function: fresnel_s (z) ¶

The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2)

The simplification fresnel_s(-x) = -fresnel_s(x) is applied when flag trigsign is true.

The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when flag %iargs is true.

See flags erf_representation and hypergeometric_representation.

‘Category: Special functions’

Option variable: erf_representation ¶

Default value: false

When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.

Option variable: hypergeometric_representation ¶

Default value: false

Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c