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

Gamma function