let-f-z-1-sin-piz-calculate-Res-f-n-with-n-integr-

Question Number 63032 by mathmax by abdo last updated on 28/Jun/19

l e t f (z) = \frac{1}{s i n (π z)} c a l c u l a t e R e s (f, n) w i t h n i n t e g r

Commented by mathmax by abdo last updated on 28/Jun/19

z = n i s s i m p l e p o l e f o r f \Rightarrow R e s (f, n) = {l i m}_{z \to n} \frac{z - n}{s i n (π z)}

c h a n g e m e n t z - n = t g i v e R e s (f, n) = {l i m}_{t \to 0} \frac{t}{s i n (π (n + t)}

= {l i m}_{t \to 0} \frac{t}{{(- 1)}^{n} s i n (π t)} = {l i m}_{t \to 0} \frac{{(- 1)}^{n}}{\frac{s i n (π t)}{π t} π} = \frac{{(- 1)}^{n}}{π}