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. This can also be expressed in terms of a hypergeometric function. See hypergeometric_representation.

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).

This can also be expressed in terms of a hypergeometric function. See hypergeometric_representation.

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):

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

This can also be expressed in terms of a hypergeometric function. See hypergeometric_representation.

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 \(C(-x) = -C(x)\) is applied when flag trigsign is true.

The simplification \(C(ix) = iC(x)\) 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 \(S(-x) = -S(x)\) is applied when flag trigsign is true.

The simplification \(S(ix) = iS(x)\) is applied when flag %iargs is true.

See flags erf_representation and hypergeometric_representation.

Categories: Special functions ·

Option variable: erf_representation ¶

Default value: false

erf_representation controls how the error functions are represented. It must be set to one of false, erf, erfc, or erfi. When set to false, the error functions are not modified. When set to erf, all error functions (erfc, erfi, erf_generalized, fresnel_s and fresnel_c) are converted to erf functions. Similary, erfc converts error functions to erfc. Finally erfi converts the functions to erfi.

Converting to erf:

(%i1) erf_representation:erf;
(%o1)                                true
(%i2) erfc(z);
(%o2)                               erfc(z)
(%i3) erfi(z);
(%o3)                               erfi(z)
(%i4) erf_generalized(z1,z2);
(%o4)                          erf(z2) - erf(z1)
(%i5) fresnel_c(z);
                    sqrt(%pi) (%i + 1) z           sqrt(%pi) (1 - %i) z
      (1 - %i) (erf(--------------------) + %i erf(--------------------))
                             2                              2
(%o5) -------------------------------------------------------------------
                                       4
(%i6) fresnel_s(z);
                    sqrt(%pi) (%i + 1) z           sqrt(%pi) (1 - %i) z
      (%i + 1) (erf(--------------------) - %i erf(--------------------))
                             2                              2
(%o6) -------------------------------------------------------------------
                                       4

Converting to erfc:

(%i1) erf_representation:erfc;
(%o1)                                erfc
(%i2) erf(z);
(%o2)                             1 - erfc(z)
(%i3) erfc(z);
(%o3)                               erfc(z)
(%i4) erf_generalized(z1,z2);
(%o4)                         erfc(z1) - erfc(z2)
(%i5) fresnel_s(c);
                         sqrt(%pi) (%i + 1) c
(%o5) ((%i + 1) ((- erfc(--------------------))
                                  2
                                                 sqrt(%pi) (1 - %i) c
                                  - %i (1 - erfc(--------------------)) + 1))/4
                                                          2
(%i6) fresnel_c(c);
                         sqrt(%pi) (%i + 1) c
(%o6) ((1 - %i) ((- erfc(--------------------))
                                  2
                                                 sqrt(%pi) (1 - %i) c
                                  + %i (1 - erfc(--------------------)) + 1))/4
                                                          2

Converting to erfc:

(%i1) erf_representation:erfi;
(%o1)                                erfi
(%i2) erf(z);
(%o2)                           - %i erfi(%i z)
(%i3) erfc(z);
(%o3)                          %i erfi(%i z) + 1
(%i4) erfi(z);
(%o4)                               erfi(z)
(%i5) erf_generalized(z1,z2);
(%o5)                   %i erfi(%i z1) - %i erfi(%i z2)
(%i6) fresnel_s(z);
                            sqrt(%pi) %i (%i + 1) z
(%o6) ((%i + 1) ((- %i erfi(-----------------------))
                                       2
                                                   sqrt(%pi) (1 - %i) %i z
                                            - erfi(-----------------------)))/4
                                                              2
(%i7) fresnel_c(z);
(%o7) 
                   sqrt(%pi) (1 - %i) %i z            sqrt(%pi) %i (%i + 1) z
    (1 - %i) (erfi(-----------------------) - %i erfi(-----------------------))
                              2                                  2
    ---------------------------------------------------------------------------
                                         4

Option variable: hypergeometric_representation ¶

Default value: false

Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c and other error functions.

(%i1) hypergeometric_representation:true;
(%o1)                                true
(%i2) fresnel_s(z);
                                                      2  4
                                     3    3  7     %pi  z    3
                 %pi hypergeometric([-], [-, -], - -------) z
                                     4    2  4       16
(%o2)            ---------------------------------------------
                                       6
(%i3) fresnel_c(z);
                                                    2  4
                                   1    1  5     %pi  z
(%o3)              hypergeometric([-], [-, -], - -------) z
                                   4    2  4       16
(%i4) erf(z);
                                        1    3      2
                      2 hypergeometric([-], [-], - z ) z
                                        2    2
(%o4)                 ----------------------------------
                                  sqrt(%pi)
(%i5) erfi(z);
                                         1    3    2
                       2 hypergeometric([-], [-], z ) z
                                         2    2
(%o5)                  --------------------------------
                                  sqrt(%pi)
(%i6) erfc(z);
                                          1    3      2
                        2 hypergeometric([-], [-], - z ) z
                                          2    2
(%o6)               1 - ----------------------------------
                                    sqrt(%pi)
(%i7) erf_generalized(z1,z2);
                        1    3       2
      2 hypergeometric([-], [-], - z2 ) z2
                        2    2
(%o7) ------------------------------------
                   sqrt(%pi)
                                                             1    3       2
                                           2 hypergeometric([-], [-], - z1 ) z1
                                                             2    2
                                         - ------------------------------------
                                                        sqrt(%pi)