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

Gauss hypergeometric function