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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\!\left(x\right)\right| \prod_{k=0}^{\infty} {\left(1 + \frac{{y}^{2}}{{\left(x + k\right)}^{2}}\right)}^{-1 / 2}
Assumptions:xRandyRandx+yi{0,1,}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\!\left(x\right)\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
Absz\left|z\right| Absolute value
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
ConstIii Imaginary unit
Productnf ⁣(n)\prod_{n} f\!\left(n\right) 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:
Entry(ID("513a30"),
    Formula(Equal(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Mul(Abs(GammaFunction(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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2019-09-22 15:43:45.410764 UTC