Fungrim entry: ca9123

\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:

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
`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
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`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("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)))))

Topics using this entry

Gauss hypergeometric function