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Fungrim entry: ca9123

sin ⁣(π(cab))π2F1 ⁣(a,b,c,z)=zaΓ ⁣(ca)Γ ⁣(cb)2F1 ⁣(a,ac+1,a+bc+1,11z)zac(1z)cabΓ ⁣(a)Γ ⁣(b)2F1 ⁣(ca,1a,cab+1,11z)\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{z}^{-a}}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a + b - c + 1, 1 - \frac{1}{z}\right) - \frac{{z}^{a - c} {\left(1 - z\right)}^{c - a - b}}{\Gamma\!\left(a\right) \Gamma\!\left(b\right)} \,{}_2{\textbf F}_1\!\left(c - a, 1 - a, c - a - b + 1, 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(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{z}^{-a}}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a + b - c + 1, 1 - \frac{1}{z}\right) - \frac{{z}^{a - c} {\left(1 - z\right)}^{c - a - b}}{\Gamma\!\left(a\right) \Gamma\!\left(b\right)} \,{}_2{\textbf F}_1\!\left(c - a, 1 - a, c - a - b + 1, 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("ca9123"),
    Formula(Equal(Mul(Div(Sin(Mul(ConstPi, Sub(Sub(c, a), b))), ConstPi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(z, Neg(a)), Mul(GammaFunction(Sub(c, a)), GammaFunction(Sub(c, b)))), Hypergeometric2F1Regularized(a, Add(Sub(a, c), 1), Add(Sub(Add(a, b), c), 1), Sub(1, Div(1, z)))), Mul(Div(Mul(Pow(z, Sub(a, c)), Pow(Sub(1, z), Sub(Sub(c, a), b))), Mul(GammaFunction(a), GammaFunction(b))), Hypergeometric2F1Regularized(Sub(c, a), Sub(1, a), Add(Sub(Sub(c, a), b), 1), Sub(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.

2019-08-21 11:44:15.926409 UTC