Error Functions¶
This module provides symbolic error functions. These functions use the \(mpmath library\) for numerical evaluation and Maxima, Pynac for symbolics.
The main objects which are exported from this module are:
erf
– The error function
erfc
– The complementary error function
erfi
– The imaginary error function
erfinv
– The inverse error function
fresnel_sin
– The Fresnel integral \(S(x)\)
fresnel_cos
– The Fresnel integral \(C(x)\)
AUTHORS:
Original authors
erf
/error_fcn
(c) 2006-2014: Karl-Dieter Crisman, Benjamin Jones, Mike Hansen, William Stein, Burcin Erocal, Jeroen Demeyer, W. D. Joyner, R. Andrew OhanaReorganisation in new file, addition of
erfi
/erfinv
/erfc
(c) 2016: Ralf StephanFresnel integrals (c) 2017 Marcelo Forets
REFERENCES:
-
class
sage.functions.error.
Function_Fresnel_cos
¶ Bases:
sage.symbolic.function.BuiltinFunction
The cosine Fresnel integral.
It is defined by the integral
\[\operatorname{C}(x) = \int_0^x \cos\left(\frac{\pi t^2}{2}\right)\, dt\]for real \(x\). Using power series expansions, it can be extended to the domain of complex numbers. See the Wikipedia article Fresnel_integral.
INPUT:
x
– the argument of the function
EXAMPLES:
sage: fresnel_cos(0) 0 sage: fresnel_cos(x).subs(x==0) 0 sage: x = var('x') sage: fresnel_cos(1).n(100) 0.77989340037682282947420641365 sage: fresnel_cos(x)._sympy_() fresnelc(x)
-
class
sage.functions.error.
Function_Fresnel_sin
¶ Bases:
sage.symbolic.function.BuiltinFunction
The sine Fresnel integral.
It is defined by the integral
\[\operatorname{S}(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right)\, dt\]for real \(x\). Using power series expansions, it can be extended to the domain of complex numbers. See the Wikipedia article Fresnel_integral.
INPUT:
x
– the argument of the function
EXAMPLES:
sage: fresnel_sin(0) 0 sage: fresnel_sin(x).subs(x==0) 0 sage: x = var('x') sage: fresnel_sin(1).n(100) 0.43825914739035476607675669662 sage: fresnel_sin(x)._sympy_() fresnels(x)
-
class
sage.functions.error.
Function_erf
¶ Bases:
sage.symbolic.function.BuiltinFunction
The error function.
The error function is defined for real values as
\[\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt.\]This function is also defined for complex values, via analytic continuation.
EXAMPLES:
We can evaluate numerically:
sage: erf(2) erf(2) sage: erf(2).n() 0.995322265018953 sage: erf(2).n(100) 0.99532226501895273416206925637 sage: erf(ComplexField(100)(2+3j)) -20.829461427614568389103088452 + 8.6873182714701631444280787545*I
Basic symbolic properties are handled by Sage and Maxima:
sage: x = var("x") sage: diff(erf(x),x) 2*e^(-x^2)/sqrt(pi) sage: integrate(erf(x),x) x*erf(x) + e^(-x^2)/sqrt(pi)
ALGORITHM:
Sage implements numerical evaluation of the error function via the
erf()
function from mpmath. Symbolics are handled by Sage and Maxima.REFERENCES:
-
class
sage.functions.error.
Function_erfc
¶ Bases:
sage.symbolic.function.BuiltinFunction
The complementary error function.
The complementary error function is defined by
\[\frac{2}{\sqrt{\pi}} \int_t^\infty e^{-x^2} dx.\]EXAMPLES:
sage: erfc(6) erfc(6) sage: erfc(6).n() 2.15197367124989e-17 sage: erfc(RealField(100)(1/2)) 0.47950012218695346231725334611 sage: 1 - erfc(0.5) 0.520499877813047 sage: erf(0.5) 0.520499877813047
-
class
sage.functions.error.
Function_erfi
¶ Bases:
sage.symbolic.function.BuiltinFunction
The imaginary error function.
The imaginary error function is defined by
\[\operatorname{erfi}(x) = -i \operatorname{erf}(ix).\]
-
class
sage.functions.error.
Function_erfinv
¶ Bases:
sage.symbolic.function.BuiltinFunction
The inverse error function.
The inverse error function is defined by:
\[\operatorname{erfinv}(x) = \operatorname{erf}^{-1}(x).\]