Fungrim entry: 90ac58

$\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b\right) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a - b + 1, \frac{1}{z}\right) - \frac{{\left(-z\right)}^{-b}}{\Gamma\!\left(a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, b - c + 1, b - a + 1, \frac{1}{z}\right)$
Assumptions:$a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left\{0, 1\right\}$
TeX:
\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b\right) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a - b + 1, \frac{1}{z}\right) - \frac{{\left(-z\right)}^{-b}}{\Gamma\!\left(a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, b - c + 1, b - a + 1, \frac{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\{0, 1\right\}
Definitions:
Fungrim symbol Notation Short description
Sin$\sin\!\left(z\right)$ Sine
ConstPi$\pi$ The constant pi (3.14...)
Hypergeometric2F1Regularized$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$ Regularized Gauss hypergeometric function
Pow${a}^{b}$ Power
GammaFunction$\Gamma\!\left(z\right)$ Gamma function
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("90ac58"),
Formula(Equal(Mul(Div(Sin(Mul(ConstPi, Sub(b, a))), ConstPi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(Neg(z), Neg(a)), Mul(GammaFunction(b), GammaFunction(Sub(c, a)))), Hypergeometric2F1Regularized(a, Add(Sub(a, c), 1), Add(Sub(a, b), 1), Div(1, z))), Mul(Div(Pow(Neg(z), Neg(b)), Mul(GammaFunction(a), GammaFunction(Sub(c, b)))), Hypergeometric2F1Regularized(b, Add(Sub(b, c), 1), Add(Sub(b, a), 1), Div(1, z)))))),
Variables(a, b, c, z),
Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, Set(0, 1)))))

Topics using this entry

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

2019-08-25 15:30:03.056001 UTC