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Fungrim entry: 3d6d7e

2F1 ⁣(a,b,c,z)=2F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) = \overline{\,{}_2F_1\!\left(\overline{a}, \overline{b}, \overline{c}, \overline{z}\right)}
Assumptions:aC  and  bC  and  cC{0,1,}  and  zC[1,)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}}\; z \in \mathbb{C} \setminus \left[1, \infty\right)
\,{}_2F_1\!\left(a, b, c, z\right) = \overline{\,{}_2F_1\!\left(\overline{a}, \overline{b}, \overline{c}, \overline{z}\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}}\; z \in \mathbb{C} \setminus \left[1, \infty\right)
Fungrim symbol Notation Short description
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Conjugatez\overline{z} Complex conjugate
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(Hypergeometric2F1(a, b, c, z), Conjugate(Hypergeometric2F1(Conjugate(a), Conjugate(b), Conjugate(c), Conjugate(z))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))

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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