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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(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, 1 - a, c - a - b + 1, 1 - \frac{1}{z}\right)
Assumptions:aC  and  bC  and  cC  and  zC  and  z(,0]  and  z[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(-\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{{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(a) \Gamma(b)} \,{}_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(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
Sinsin(z)\sin(z) Sine
Piπ\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
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Source code for this entry:
Entry(ID("ca9123"),
    Formula(Equal(Mul(Div(Sin(Mul(Pi, Sub(Sub(c, a), b))), Pi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(z, Neg(a)), Mul(Gamma(Sub(c, a)), Gamma(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(Gamma(a), Gamma(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, OpenClosedInterval(Neg(Infinity), 0)), NotElement(z, ClosedOpenInterval(1, Infinity)))))

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

2020-04-08 16:14:44.404316 UTC