# Fungrim entry: 48910b

$\mathrm{B}\!\left(a, b\right) = 2 \int_{0}^{\pi / 2} \sin^{2 a - 1}\!\left(t\right) \cos^{2 b - 1}\!\left(t\right) \, dt$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0$
TeX:
\mathrm{B}\!\left(a, b\right) = 2 \int_{0}^{\pi / 2} \sin^{2 a - 1}\!\left(t\right) \cos^{2 b - 1}\!\left(t\right) \, dt

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
BetaFunction$\mathrm{B}\!\left(a, b\right)$ Beta function
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Sin$\sin(z)$ Sine
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("48910b"),
Formula(Equal(BetaFunction(a, b), Mul(2, Integral(Mul(Pow(Sin(t), Sub(Mul(2, a), 1)), Pow(Cos(t), Sub(Mul(2, b), 1))), For(t, 0, Div(Pi, 2)))))),
Variables(a, b),
Assumptions(And(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.

2020-03-29 16:01:42.585089 UTC