Fungrim entry: bf533f

\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

References:

https://doi.org/10.1080%2F10652469.2017.1376193

TeX:

\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Gamma`	$\Gamma(z)$	Gamma function
`Exp`	${e}^{z}$	Exponential function
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Product`	$\prod_{n} f(n)$	Product
`DigammaFunctionZero`	$x_{n}$	Zero of the digamma function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("bf533f"),
    Formula(Equal(Div(DigammaFunction(z), Gamma(z)), Neg(Mul(Exp(Mul(Mul(2, ConstGamma), z)), Product(Mul(Sub(1, Div(z, DigammaFunctionZero(n))), Exp(Div(z, DigammaFunctionZero(n)))), For(n, 0, Infinity)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))),
    References("https://doi.org/10.1080%2F10652469.2017.1376193"))

Topics using this entry

Digamma function