# Fungrim entry: a1941b

$I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt$
Assumptions:$x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0$
TeX:
I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0
Definitions:
Fungrim symbol Notation Short description
IncompleteBetaRegularized$I_{x}\!\left(a, b\right)$ Regularized incomplete beta function
BetaFunction$\mathrm{B}\!\left(a, b\right)$ Beta function
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
ClosedInterval$\left[a, b\right]$ Closed interval
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("a1941b"),
Formula(Equal(IncompleteBetaRegularized(x, a, b), Mul(Div(1, BetaFunction(a, b)), Integral(Mul(Pow(t, Sub(a, 1)), Pow(Sub(1, t), Sub(b, 1))), For(t, 0, x))))),
Variables(x, a, b),
Assumptions(And(Element(x, ClosedInterval(0, 1)), Element(a, CC), Element(b, CC), Greater(Re(a), 0), Greater(Re(b), 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