Fungrim entry: 513a30

\left|\Gamma\!\left(x + y i\right)\right| = \left|\Gamma(x)\right| \prod_{k=0}^{\infty} {\left(1 + \frac{{y}^{2}}{{\left(x + k\right)}^{2}}\right)}^{-1 / 2}

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \{0, -1, \ldots\}

References:

Abramowitz & Stegun 6.1.25

TeX:

\left|\Gamma\!\left(x + y i\right)\right| = \left|\Gamma(x)\right| \prod_{k=0}^{\infty} {\left(1 + \frac{{y}^{2}}{{\left(x + k\right)}^{2}}\right)}^{-1 / 2}

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Gamma`	$\Gamma(z)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("513a30"),
    Formula(Equal(Abs(Gamma(Add(x, Mul(y, ConstI)))), Mul(Abs(Gamma(x)), Product(Pow(Add(1, Div(Pow(y, 2), Pow(Add(x, k), 2))), Neg(Div(1, 2))), For(k, 0, Infinity))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR), NotElement(Add(x, Mul(y, ConstI)), ZZLessEqual(0)))),
    References("Abramowitz & Stegun 6.1.25"))

Topics using this entry

Bounds and inequalities for the gamma function