let-f-z-sin-z-z-2-calculate-Res-f-0-

Question Number 63031 by mathmax by abdo last updated on 28/Jun/19

l e t f (z) = \frac{s i n (z)}{z^{2}} c a l c u l a t e R e s (f, 0)

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

0 i s a d o u b l e p o l e \Rightarrow R e s (f, 0) = {l i m}_{z \to 0} \frac{1}{(2 - 1)!} {z^{2} f (z)}^{(1)}

= {l i m}_{z \to 0} {s i n z}^{(1)} = {l i m}_{z \to 0} c o s z = 1

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

a n o t h e r w a y i f f (z) = \sum_{k = 0}^{\infty} a_{k} {(z - z_{0})}^{k} + \sum_{k = 1}^{\infty} a_{- k} {(z - z_{0})}^{- k} (t b e l a u r e n t s e r i e o f f (z))

\to R e s (f, z_{0}) = a_{- 1} = c o e f f i c i e n t o f \frac{1}{z - z_{o}} i n t h i s s e r i e s o w e h a v e

s i n (z) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} z^{2 n + 1}}{(2 n + 1)!} = z - \frac{z^{3}}{3!} + \frac{z^{5}}{5!} - \dots . \Rightarrow

\frac{s i n z}{z^{2}} = \frac{1}{z} - \frac{z}{3!} + \frac{z^{3}}{5!} - \dots . \Rightarrow R e (f, 0) = 1 (c o e f f i c e n t o f \frac{1}{z}) .