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Fungrim entry: 5ec9c0

Bx ⁣(a,b)=xaa2F1 ⁣(a,1b,a+1,x)\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)
Assumptions:aC{0,1,}andbCandxC{1}and(x0orRe ⁣(a)>0)a \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \setminus \left\{1\right\} \,\mathbin{\operatorname{and}}\, \left(x \ne 0 \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(a\right) > 0\right)
TeX:
\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \setminus \left\{1\right\} \,\mathbin{\operatorname{and}}\, \left(x \ne 0 \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(a\right) > 0\right)
Definitions:
Fungrim symbol Notation Short description
IncompleteBetaBx ⁣(a,b)\mathrm{B}_{x}\!\left(a, b\right) Incomplete beta function
Powab{a}^{b} Power
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("5ec9c0"),
    Formula(Equal(IncompleteBeta(x, a, b), Mul(Div(Pow(x, a), a), Hypergeometric2F1(a, Sub(1, b), Add(a, 1), x)))),
    Variables(x, a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, CC), Element(x, SetMinus(CC, Set(1))), Or(Unequal(x, 0), Greater(Re(a), 0)))))

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

2019-08-25 15:30:03.056001 UTC