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Fungrim entry: cb93ea

erf ⁣(z)=zz2ez2zπU ⁣(12,12,z2)\operatorname{erf}\!\left(z\right) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)
Assumptions:zCandz0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0
TeX:
\operatorname{erf}\!\left(z\right) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0
Definitions:
Fungrim symbol Notation Short description
Erferf ⁣(z)\operatorname{erf}\!\left(z\right) Error function
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("cb93ea"),
    Formula(Equal(Erf(z), Sub(Div(z, Sqrt(Pow(z, 2))), Mul(Div(Exp(Neg(Pow(z, 2))), Mul(z, Sqrt(ConstPi))), HypergeometricUStar(Div(1, 2), Div(1, 2), Pow(z, 2)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Unequal(z, 0))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-15 11:00:55.020619 UTC