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Fungrim entry: f7f84e

1F1 ⁣(a,b,z)=(z)aΓ ⁣(ba)U ⁣(a,b,z)+zabezΓ(a)U ⁣(ba,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma(a)} U^{*}\!\left(b - a, b, -z\right)
Assumptions:aC  and  bC  and  zC  and  z0a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0
TeX:
\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma(a)} U^{*}\!\left(b - a, b, -z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0
Definitions:
Fungrim symbol Notation Short description
Hypergeometric1F1Regularized1F1 ⁣(a,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) Regularized Kummer confluent hypergeometric function
Powab{a}^{b} Power
GammaΓ(z)\Gamma(z) Gamma function
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("f7f84e"),
    Formula(Equal(Hypergeometric1F1Regularized(a, b, z), Add(Mul(Div(Pow(Neg(z), Neg(a)), Gamma(Sub(b, a))), HypergeometricUStar(a, b, z)), Mul(Div(Mul(Pow(z, Sub(a, b)), Exp(z)), Gamma(a)), HypergeometricUStar(Sub(b, a), b, Neg(z)))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), NotEqual(z, 0))))

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

2020-04-08 16:14:44.404316 UTC