Fungrim home page

Fungrim entry: 659ce8

2F1 ⁣(a,b,c,1)=Γ(c)Γ ⁣(cab)Γ ⁣(ca)Γ ⁣(cb)\,{}_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:aC  and  bC  and  cC{0,1,}  and  Re ⁣(cab)>0a \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
\,{}_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
Fungrim symbol Notation Short description
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    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