15.6 Error Function

The Error function and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 7 and (DLMF 7)

Function: erf (z) ¶

The Error Function erf(z):

\[{\rm erf}\ z = {{2\over \sqrt{\pi}}} \int_0^z e^{-t^2}\, dt \]

(A&S eqn 7.1.1) and (DLMF 7.2.E1).

See also flag erfflag.

Categories: Special functions ·

Function: erfc (z) ¶

The Complementary Error Function erfc(z):

\[{\rm erfc}\ z = 1 - {\rm erf}\ z \]

(A&S eqn 7.1.2) and (DLMF 7.2.E2).

Categories: Special functions ·

Function: erfi (z) ¶

The Imaginary Error Function.

\[{\rm erfi}\ z = -i\, {\rm erf}(i z) \]

Categories: Special functions ·

Function: erf_generalized (z1,z2) ¶

Generalized Error function Erf(z1,z2):

\[{\rm erf}(z_1, z_2) = {{2\over \sqrt{\pi}}} \int_{z_1}^{z_2} e^{-t^2}\, dt \]

Categories: Special functions ·

Function: fresnel_c (z) ¶

The Fresnel Integral

\[C(z) = \int_0^z \cos\left({\pi \over 2} t^2\right)\, dt \]

(A&S eqn 7.3.1) and (DLMF 7.2.E7).

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.

Categories: Special functions ·

Function: fresnel_s (z) ¶

The Fresnel Integral

\[S(z) = \int_0^z \sin\left({\pi \over 2} t^2\right)\, dt \]

(A&S eqn 7.3.2) and (DLMF 7.2.E8).

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.

Categories: Special functions ·

Option variable: erf_representation ¶

Default value: false

When true, 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.