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Fungrim entry: 1f72e9

resz=aB ⁣(z,b)={(nbn),nZ00,otherwise   where n=a\mathop{\operatorname{res}}\limits_{z=a} \mathrm{B}\!\left(z, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a
Assumptions:aCandbC{0,1,}a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\}
TeX:
\mathop{\operatorname{res}}\limits_{z=a} \mathrm{B}\!\left(z, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
Residueresz=cf(z)\mathop{\operatorname{res}}\limits_{z=c} f(z) Complex residue
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
Binomial(nk){n \choose k} Binomial coefficient
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("1f72e9"),
    Formula(Equal(Residue(BetaFunction(z, b), For(z, a)), Where(Cases(Tuple(Binomial(Sub(n, b), n), Element(n, ZZGreaterEqual(0))), Tuple(0, Otherwise)), Equal(n, Neg(a))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, SetMinus(CC, 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.

2019-10-05 13:11:19.856591 UTC