let-z-from-C-Z-prove-that-pi-sin-piz-1-z-n-1-1-n-2z-z-2-n-2-and-picos-piz-sin-piz-1-z-n-1-2z-z-2-n-2-

Tinku Tara June 3, 2023 Relation and Functions 0 Comments

Question Number 67522 by mathmax by abdo last updated on 28/Aug/19

let z from C−Z prove that (π/(sin(πz))) =(1/z) +Σ_(n=1) ^∞ (((−1)^n 2z)/(z^2 −n^2 )) and ((πcos(πz))/(sin(πz))) =(1/z) +Σ_(n=1) ^∞ ((2z)/(z^2 −n^2 ))

$${let}\:{z}\:{from}\:{C}−{Z}\:\:\:\:\:{prove}\:{that} \\ $$$$\frac{\pi}{{sin}\left(\pi{z}\right)}\:=\frac{\mathrm{1}}{{z}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \mathrm{2}{z}}{{z}^{\mathrm{2}} −{n}^{\mathrm{2}} }\:\:{and} \\ $$$$\frac{\pi{cos}\left(\pi{z}\right)}{{sin}\left(\pi{z}\right)}\:=\frac{\mathrm{1}}{{z}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{2}{z}}{{z}^{\mathrm{2}} −{n}^{\mathrm{2}} } \\ $$

Facebook Tweet Pin

let-z-from-C-Z-prove-that-pi-sin-piz-1-z-n-1-1-n-2z-z-2-n-2-and-picos-piz-sin-piz-1-z-n-1-2z-z-2-n-2-

Leave a Reply Cancel reply