find-residue-f-z-z-3-csc-z-2-

Question Number 145810 by tabata last updated on 08/Jul/21

f i n d r e s i d u e f (z) = z^{- 3} c s c (z^{2})

Commented by tabata last updated on 08/Jul/21

h o w c a n i t s o l v e t h i s

Commented by tabata last updated on 08/Jul/21

? ? ? ? ?

Answered by Olaf_Thorendsen last updated on 09/Jul/21

f (z) = z^{- 3} \csc (z^{2}) = \frac{1}{z^{3} \sin (z^{2})}

The value z = 0 is a pole of f .

f (z) \underset{0}{\sim} \frac{1}{z^{3} \times (z^{2} - \frac{z^{6}}{3!} + \frac{z^{10}}{5!} \dots)}

f (z) \underset{0}{\sim} \frac{1}{z^{5} (1 - \frac{z^{4}}{3!} + \frac{z^{8}}{5!} \dots)}

f (z) \underset{0}{\sim} \frac{1}{z^{5}} (1 + \frac{z^{4}}{3!} - \frac{z^{8}}{5!} + \dots)

The coef of \frac{1}{z} is \frac{1}{3!} = \frac{1}{6}

\Rightarrow {Res}_{0} f = \frac{1}{6}