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Fungrim entry: 6f3fec

Im⁡ ⁣(ψ ⁣(12+iy))=π2tanh⁡ ⁣(πy)\operatorname{Im}\!\left(\psi\!\left(\frac{1}{2} + i y\right)\right) = \frac{\pi}{2} \tanh\!\left(\pi y\right)
Assumptions:y∈R  and⁡  y≠0y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0
TeX:
\operatorname{Im}\!\left(\psi\!\left(\frac{1}{2} + i y\right)\right) = \frac{\pi}{2} \tanh\!\left(\pi y\right)

y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \ne 0
Definitions:
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:
Entry(ID("6f3fec"),
    Formula(Equal(Im(DigammaFunction(Add(Div(1, 2), Mul(ConstI, y)))), Mul(Div(Pi, 2), Tanh(Mul(Pi, y))))),
    Variables(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