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Fungrim entry: 6430cc

eπz=π(1Γ ⁣(12+iz)Γ ⁣(12iz)+zΓ ⁣(1+iz)Γ ⁣(1iz)){e}^{\pi z} = \pi \left(\frac{1}{\Gamma\!\left(\frac{1}{2} + i z\right) \Gamma\!\left(\frac{1}{2} - i z\right)} + \frac{z}{\Gamma\!\left(1 + i z\right) \Gamma\!\left(1 - i z\right)}\right)
Assumptions:zCz \in \mathbb{C}
TeX:
{e}^{\pi z} = \pi \left(\frac{1}{\Gamma\!\left(\frac{1}{2} + i z\right) \Gamma\!\left(\frac{1}{2} - i z\right)} + \frac{z}{\Gamma\!\left(1 + i z\right) \Gamma\!\left(1 - i z\right)}\right)

z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
GammaΓ(z)\Gamma(z) Gamma function
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("6430cc"),
    Formula(Equal(Exp(Mul(Pi, z)), Mul(Pi, Add(Div(1, Mul(Gamma(Add(Div(1, 2), Mul(ConstI, z))), Gamma(Sub(Div(1, 2), Mul(ConstI, z))))), Div(z, Mul(Gamma(Add(1, Mul(ConstI, z))), Gamma(Sub(1, Mul(ConstI, z))))))))),
    Variables(z),
    Assumptions(Element(z, CC)))

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2021-03-15 19:12:00.328586 UTC