# 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

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

2021-03-15 19:12:00.328586 UTC