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The Error function and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 7 and (DLMF 7)
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.
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.
The Imaginary Error Function. $$ {\rm erfi}\ z = -i\, {\rm erf}(i z) $$
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.
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.
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.
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
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)
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