find-residu-of-function-f-z-e-1-z-z-2-1-in-z-0-

Question Number 56145 by gunawan last updated on 11/Mar/19

find residu of function

f (z) = \frac{e^{\frac{1}{z}}}{z^{2} + 1} in z = 0

Answered by einsteindrmaths@hotmail.fr last updated on 05/Apr/19

i f f (z) = Σ a_{n} {(z - a)}^{n} r e s (f . a) = a_{- 1}

e (1 / z) = Σ \frac{1}{n! z^{n}}

\frac{1}{z^{2} + 1} = Σ {(- z^{2})}^{n}

f (z) = Σ {(- 1)}^{m} {(z)}^{2 m} . Σ \frac{1}{n! z^{n}}

a_{- 1} = \sum_{2 m - n = - 1} {(- 1)}^{m} \frac{1}{n!} = \sum_{m} {(- 1)}^{m} . \frac{1}{(2 m + 1)!} = \sin (1)