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

Beta function