Fungrim entry: 659ce8

\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma\!\left(c\right) \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) \gt 0

TeX:

\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma\!\left(c\right) \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) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("659ce8"),
    Formula(Equal(Hypergeometric2F1(a, b, c, 1), Div(Mul(GammaFunction(c), GammaFunction(Sub(Sub(c, a), b))), Mul(GammaFunction(Sub(c, a)), GammaFunction(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

Gauss hypergeometric function