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Fungrim entry: 90ac58

sin ⁣(π(ba))π2F1 ⁣(a,b,c,z)=(z)aΓ ⁣(b)Γ ⁣(ca)2F1 ⁣(a,ac+1,ab+1,1z)(z)bΓ ⁣(a)Γ ⁣(cb)2F1 ⁣(b,bc+1,ba+1,1z)\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:aCandbCandcCandzCandz{0,1}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
Sinsin ⁣(z)\sin\!\left(z\right) Sine
ConstPiπ\pi The constant pi (3.14...)
Hypergeometric2F1Regularized2F1 ⁣(a,b,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function
Powab{a}^{b} Power
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
CCC\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)))))

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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