Assumptions:
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 | Beta function | |
Integral | Integral | |
Pow | Power | |
Sin | Sine | |
Cos | Cosine | |
Pi | The constant pi (3.14...) | |
CC | Complex numbers | |
Re | 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))))