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


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Question Number 198175 by York12 last updated on 12/Oct/23
$Prove The following Functional equation: ζ(x,s)=((2Γ(1−s))/((2π)^((1−s)) )){sin(((πs)/2))Σ_(m=1) ^∞ [((cos(2πmx))/m^((1−s)) )]+cos(((πs)/2))Σ_(m=1) ^∞ [((sin(2πmx))/m^((1−s)) )]}$
$P r o v e T h e f o l l o w i n g F u n c t i o n a l e q u a t i o n :$ $ζ (x, s) = \frac{2 Γ (1 - s)}{{(2 π)}^{(1 - s)}} {s i n (\frac{π s}{2}) \underset{m = 1}{\sum^{\infty}} [\frac{c o s (2 π m x)}{m^{(1 - s)}}] + c o s (\frac{π s}{2}) \underset{m = 1}{\sum^{\infty}} [\frac{s i n (2 π m x)}{m^{(1 - s)}}]}$
	Answered by witcher3 last updated on 13/Oct/23

	$google$ $poissant sumation$ $Σ f (n) = Σ \int_{0}^{\infty} e^{- 2 i π nt} f (t) dt . . . .$
	Commented by York12 last updated on 13/Oct/23

	$T h a n k s$
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