# Fungrim entry: 2df3e3

$\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 a \notin \{1, 0, \ldots\}$
TeX:
\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 a \notin \{1, 0, \ldots\}
Definitions:
Fungrim symbol Notation Short description
Hypergeometric0F1$\,{}_0F_1\!\left(a, z\right)$ Confluent hypergeometric limit function
Exp${e}^{z}$ Exponential function
Sqrt$\sqrt{z}$ Principal square root
Hypergeometric1F1$\,{}_1F_1\!\left(a, b, z\right)$ Kummer confluent hypergeometric function
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("2df3e3"),
Formula(Equal(Hypergeometric0F1(a, z), Mul(Exp(Neg(Mul(2, Sqrt(z)))), Hypergeometric1F1(Sub(a, Div(1, 2)), Sub(Mul(2, a), 1), Mul(4, Sqrt(z)))))),
Variables(a, z),
Assumptions(And(Element(a, CC), Element(z, CC), NotElement(Mul(2, a), ZZLessEqual(1)))))

## 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