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Fungrim entry: 513a30

Γ ⁣(x+yi)=Γ(x)k=0(1+y2(x+k)2)1/2\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:xR  and  yR  and  x+yi{0,1,}x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \{0, -1, \ldots\}
  • Abramowitz & Stegun 6.1.25
\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\}
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
GammaΓ(z)\Gamma(z) Gamma function
ConstIii Imaginary unit
Productnf(n)\prod_{n} f(n) Product
Powab{a}^{b} Power
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    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"))

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2021-03-15 19:12:00.328586 UTC