Fungrim entry: 5ec9c0

\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(a) > 0\right)

TeX:

\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(a) > 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`Pow`	${a}^{b}$	Power
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("5ec9c0"),
    Formula(Equal(IncompleteBeta(x, a, b), Mul(Div(Pow(x, a), a), Hypergeometric2F1(a, Sub(1, b), Add(a, 1), x)))),
    Variables(x, a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, CC), Element(x, SetMinus(CC, Set(1))), Or(NotEqual(x, 0), Greater(Re(a), 0)))))

Topics using this entry

Beta function