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Fungrim entry: ae3110

erfc ⁣(z)=ez2zπU ⁣(12,12,z2)\operatorname{erfc}\!\left(z\right) = \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)
Assumptions:zCandRe ⁣(z)>0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) > 0
TeX:
\operatorname{erfc}\!\left(z\right) = \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}}\, \operatorname{Re}\!\left(z\right) > 0
Definitions:
Fungrim symbol Notation Short description
Erfcerfc ⁣(z)\operatorname{erfc}\!\left(z\right) Complementary error function
Expez{e}^{z} Exponential function
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
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
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("ae3110"),
    Formula(Equal(Erfc(z), 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), Greater(Re(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 14:14:26.267625 UTC