# Fungrim entry: 6430cc

${e}^{\pi z} = \pi \left(\frac{1}{\Gamma\!\left(\frac{1}{2} + i z\right) \Gamma\!\left(\frac{1}{2} - i z\right)} + \frac{z}{\Gamma\!\left(1 + i z\right) \Gamma\!\left(1 - i z\right)}\right)$
Assumptions:$z \in \mathbb{C}$
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
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
Gamma$\Gamma(z)$ Gamma function
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ 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)))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC