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Lambert-series-type-representation-for-1-factorial-i-i-i-i-1-2-n-1-1-n-e-n-2-1-




Question Number 163548 by HongKing last updated on 07/Jan/22
Lambert series type representation  for  (√(-1))  factorial  (i^i )! (-i^i )! = 1 - 2 Σ_(n=1) ^∞  (((-1)^n )/(e^𝛑  n^2  - 1))
$$\mathrm{Lambert}\:\mathrm{series}\:\mathrm{type}\:\mathrm{representation} \\ $$$$\mathrm{for}\:\:\sqrt{-\mathrm{1}}\:\:\mathrm{factorial} \\ $$$$\left(\mathrm{i}^{\boldsymbol{\mathrm{i}}} \right)!\:\left(-\mathrm{i}^{\boldsymbol{\mathrm{i}}} \right)!\:=\:\mathrm{1}\:-\:\mathrm{2}\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} }{\mathrm{e}^{\boldsymbol{\pi}} \:\mathrm{n}^{\mathrm{2}} \:-\:\mathrm{1}} \\ $$

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