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Fungrim entry: 3e08b6

Bx ⁣(a,b)=0xta1(1t)b1dt\mathrm{B}_{x}\!\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}\!\left(a\right) > 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(b\right) > 0
TeX:
\mathrm{B}_{x}\!\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}\!\left(a\right) > 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(b\right) > 0
Definitions:
Fungrim symbol Notation Short description
IncompleteBetaBx ⁣(a,b)\mathrm{B}_{x}\!\left(a, b\right) Incomplete beta function
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, dx Integral
Powab{a}^{b} Power
ClosedInterval[a,b]\left[a, b\right] Closed interval
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("3e08b6"),
    Formula(Equal(IncompleteBeta(x, a, b), Integral(Mul(Pow(t, Sub(a, 1)), Pow(Sub(1, t), Sub(b, 1))), Tuple(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.

2019-08-25 15:30:03.056001 UTC