Fungrim home page

Fungrim entry: 22a9cd

Im ⁣(ψ ⁣(1+iy))=π2coth ⁣(πy)12y\operatorname{Im}\!\left(\psi\!\left(1 + i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) - \frac{1}{2 y}
Assumptions:yR  and  y0y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0
\operatorname{Im}\!\left(\psi\!\left(1 + i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) - \frac{1}{2 y}

y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0
Fungrim symbol Notation Short description
ImIm(z)\operatorname{Im}(z) Imaginary part
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ConstIii Imaginary unit
Piπ\pi The constant pi (3.14...)
RRR\mathbb{R} Real numbers
Source code for this entry:
    Formula(Equal(Im(DigammaFunction(Add(1, Mul(ConstI, y)))), Sub(Mul(Div(Pi, 2), Coth(Mul(Pi, y))), Div(1, Mul(2, y))))),
    Assumptions(And(Element(y, RR), NotEqual(y, 0))))

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