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Fungrim entry: c92da4

Ix ⁣(a,b)=Bx ⁣(a,b)B ⁣(a,b)I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\right)}
Assumptions:xC  and  aC{0,1,}  and  bC{0,1,}  and  a+b{0,1,}x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}
I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\right)}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}
Fungrim symbol Notation Short description
IncompleteBetaRegularizedIx ⁣(a,b)I_{x}\!\left(a, b\right) Regularized incomplete beta function
IncompleteBetaBx ⁣(a,b)\mathrm{B}_{x}\!\left(a, b\right) Incomplete beta function
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Div(IncompleteBeta(x, a, b), BetaFunction(a, b)))),
    Variables(x, a, b),
    Assumptions(And(Element(x, CC), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)))))

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

2021-03-15 19:12:00.328586 UTC