# Fungrim entry: 18ef23

$U\!\left(a, n, z\right) = \lim_{b \to n} \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma(a)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0$
TeX:
U\!\left(a, n, z\right) = \lim_{b \to n} \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma(a)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0
Definitions:
Fungrim symbol Notation Short description
HypergeometricU$U\!\left(a, b, z\right)$ Tricomi confluent hypergeometric function
ComplexLimit$\lim_{z \to a} f(z)$ Limiting value, complex variable
Gamma$\Gamma(z)$ Gamma function
Hypergeometric1F1$\,{}_1F_1\!\left(a, b, z\right)$ Kummer confluent hypergeometric function
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("18ef23"),
Formula(Equal(HypergeometricU(a, n, z), ComplexLimit(Add(Mul(Div(Gamma(Sub(1, b)), Gamma(Add(Sub(a, b), 1))), Hypergeometric1F1(a, b, z)), Mul(Mul(Div(Gamma(Sub(b, 1)), Gamma(a)), Pow(z, Sub(1, b))), Hypergeometric1F1(Add(Sub(a, b), 1), Sub(2, b), z))), For(b, n)))),
Variables(a, n, z),
Assumptions(And(Element(a, CC), Element(n, ZZ), 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