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Fungrim entry: bf533f

ψ ⁣(z)Γ(z)=e2γzn=0(1zxn)exp ⁣(zxn)\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:zC  and  z{0,1,}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
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
GammaΓ(z)\Gamma(z) Gamma function
Expez{e}^{z} Exponential function
ConstGammaγ\gamma The constant gamma (0.577...)
Productnf(n)\prod_{n} f(n) Product
DigammaFunctionZeroxnx_{n} Zero of the digamma function
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\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"))

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2021-03-15 19:12:00.328586 UTC