Fungrim entry: 27bc34

\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(1 - z\right)}^{-a}}{\Gamma\!\left(b\right) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, c - b, a - b + 1, \frac{1}{1 - z}\right) - \frac{{\left(1 - z\right)}^{-b}}{\Gamma\!\left(a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, c - a, b - a + 1, \frac{1}{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(1 - z\right)}^{-a}}{\Gamma\!\left(b\right) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, c - b, a - b + 1, \frac{1}{1 - z}\right) - \frac{{\left(1 - z\right)}^{-b}}{\Gamma\!\left(a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, c - a, b - a + 1, \frac{1}{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("27bc34"),
    Formula(Equal(Mul(Div(Sin(Mul(ConstPi, Sub(b, a))), ConstPi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(Sub(1, z), Neg(a)), Mul(GammaFunction(b), GammaFunction(Sub(c, a)))), Hypergeometric2F1Regularized(a, Sub(c, b), Add(Sub(a, b), 1), Div(1, Sub(1, z)))), Mul(Div(Pow(Sub(1, z), Neg(b)), Mul(GammaFunction(a), GammaFunction(Sub(c, b)))), Hypergeometric2F1Regularized(b, Sub(c, a), Add(Sub(b, a), 1), Div(1, Sub(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

Gauss hypergeometric function