# Fungrim entry: db3eb9

$\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{1}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, b, a + b - c + 1, 1 - z\right) - \frac{{\left(1 - z\right)}^{c - a - b}}{\Gamma(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c - a - b + 1, 1 - z\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)$
TeX:
\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{1}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, b, a + b - c + 1, 1 - z\right) - \frac{{\left(1 - z\right)}^{c - a - b}}{\Gamma(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c - a - b + 1, 1 - z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
Sin$\sin(z)$ Sine
Pi$\pi$ The constant pi (3.14...)
Hypergeometric2F1Regularized$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$ Regularized Gauss hypergeometric function
Gamma$\Gamma(z)$ Gamma function
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Source code for this entry:
Entry(ID("db3eb9"),
Formula(Equal(Mul(Div(Sin(Mul(Pi, Sub(Sub(c, a), b))), Pi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(1, Mul(Gamma(Sub(c, a)), Gamma(Sub(c, b)))), Hypergeometric2F1Regularized(a, b, Add(Sub(Add(a, b), c), 1), Sub(1, z))), Mul(Div(Pow(Sub(1, z), Sub(Sub(c, a), b)), Mul(Gamma(a), Gamma(b))), Hypergeometric2F1Regularized(Sub(c, a), Sub(c, b), Add(Sub(Sub(c, a), b), 1), Sub(1, z)))))),
Variables(a, b, c, z),
Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), NotElement(z, 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.

2021-03-15 19:12:00.328586 UTC