# Fungrim entry: 659ce8

$\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma(c) \Gamma\!\left(c - a - b\right)}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(c - a - b\right) > 0$
TeX:
\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma(c) \Gamma\!\left(c - a - b\right)}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(c - a - b\right) > 0
Definitions:
Fungrim symbol Notation Short description
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
Gamma$\Gamma(z)$ Gamma function
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("659ce8"),
Formula(Equal(Hypergeometric2F1(a, b, c, 1), Div(Mul(Gamma(c), Gamma(Sub(Sub(c, a), b))), Mul(Gamma(Sub(c, a)), Gamma(Sub(c, b)))))),
Variables(a, b, c),
Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Greater(Re(Sub(Sub(c, a), b)), 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