Question Number 198175 by York12 last updated on 12/Oct/23
![Prove The following Functional equation: ζ(x,s)=((2Γ(1−s))/((2π)^((1−s)) )){sin(((πs)/2))Σ_(m=1) ^∞ [((cos(2πmx))/m^((1−s)) )]+cos(((πs)/2))Σ_(m=1) ^∞ [((sin(2πmx))/m^((1−s)) )]}](https://www.tinkutara.com/question/Q198175.png)
$${Prove}\:{The}\:{following}\:{Functional}\:{equation}: \\ $$$$\zeta\left({x},{s}\right)=\frac{\mathrm{2}\Gamma\left(\mathrm{1}−{s}\right)}{\left(\mathrm{2}\pi\right)^{\left(\mathrm{1}−{s}\right)} }\left\{{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{cos}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]+{cos}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{sin}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]\right\} \\ $$
Answered by witcher3 last updated on 13/Oct/23
$$\mathrm{google} \\ $$$$\mathrm{poissant}\:\mathrm{sumation} \\ $$$$\Sigma\mathrm{f}\left(\mathrm{n}\right)=\Sigma\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{2}\boldsymbol{\mathrm{i}\pi}\mathrm{nt}} \mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}…. \\ $$
Commented by York12 last updated on 13/Oct/23
$${Thanks} \\ $$