# Fungrim entry: 1f72e9

$\mathop{\operatorname{res}}\limits_{z=a} \mathrm{B}\!\left(z, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}$
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
Residue$\mathop{\operatorname{res}}\limits_{z=c} f(z)$ Complex residue
BetaFunction$\mathrm{B}\!\left(a, b\right)$ Beta function
Binomial${n \choose k}$ Binomial coefficient
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ 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))))))

## Topics using this entry

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

2020-03-29 16:01:42.585089 UTC