let-f-z-z-2-1-z-4-1-find-a-k-the-poles-of-f-and-calculate-Res-f-a-k-

Question Number 36198 by prof Abdo imad last updated on 30/May/18

l e t f (z) = \frac{z^{2} + 1}{z^{4} - 1}

f i n d (a_{k)} t h e p o l e s o f f a n d c a l c u l a t e

R e s (f, a_{k})

Commented by prof Abdo imad last updated on 01/Jun/18

w e h a v e f (z) = \frac{z^{2} + 1}{(z^{2} - 1) (z^{2} + 1)}

= \frac{z^{2} + 1}{(z - 1) (z + 1) (z - i) (z + i)} s o t h e p o l e s o f f

a r e 1, - 1, i, - i a n d

R e s (f, 1) = {l i m}_{z \to 1} (z - 1) φ (z) = \frac{2}{2 (2)} = \frac{1}{2}

R e s (f, - 1) = {l i m}_{z \to - 1} (z + 1) φ (z)

= \frac{2}{(- 2) . 2} = - \frac{1}{2}

R e s (f, i) = {l i m}_{z \to i} (z - i) φ (z) = 0

R e s (φ, - i) = {l i m}_{z \to - i} (z + i) φ (z) = 0