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Fungrim entry: a1941b

Ix ⁣(a,b)=1B ⁣(a,b)0xta1(1t)b1dtI_{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[0,1]andaCandbCandRe(a)>0andRe(b)>0x \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
IncompleteBetaRegularizedIx ⁣(a,b)I_{x}\!\left(a, b\right) Regularized incomplete beta function
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
ClosedInterval[a,b]\left[a, b\right] Closed interval
CCC\mathbb{C} Complex numbers
ReRe(z)\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))))

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

2019-11-11 15:50:15.016492 UTC