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Fungrim entry: 48910b

B ⁣(a,b)=20π/2sin2a1 ⁣(t)cos2b1 ⁣(t)dt\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:aC  and  bC  and  Re(a)>0  and  Re(b)>0a \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
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Sinsin(z)\sin(z) Sine
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ReRe(z)\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))))

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2020-03-29 16:01:42.585089 UTC