# Fungrim entry: 3d6d7e

$\,{}_2F_1\!\left(a, b, c, z\right) = \overline{\,{}_2F_1\!\left(\overline{a}, \overline{b}, \overline{c}, \overline{z}\right)}$
Assumptions:$a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left[1, \infty\right)$
TeX:
\,{}_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)
Definitions:
Fungrim symbol Notation Short description
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
Conjugate$\overline{z}$ Complex conjugate
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("3d6d7e"),
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))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-12-30 15:00:46.909060 UTC