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:
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 Ohana- Reorganisation in new file, addition of
erfi
/erfinv
/erfc
(c) 2016: Ralf Stephan
REFERENCES:
-
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:
\[\operatorname{erfinv}(x) = \operatorname{erf}^{-1}(x).\]