Fungrim entry: f7f84e

$\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma(a)} U^{*}\!\left(b - a, b, -z\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0$
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
Hypergeometric1F1Regularized$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$ Regularized Kummer confluent hypergeometric function
Pow${a}^{b}$ Power
Gamma$\Gamma(z)$ Gamma function
HypergeometricUStar$U^{*}\!\left(a, b, z\right)$ Scaled Tricomi confluent hypergeometric function
Exp${e}^{z}$ Exponential function
CC$\mathbb{C}$ 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))))

Topics using this entry

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

2021-03-15 19:12:00.328586 UTC