let-f-z-1-z-2-e-2z-z-3-calculate-Res-f-0-

Question Number 37300 by math khazana by abdo last updated on 11/Jun/18

l e t f (z) = \frac{(1 - z^{2}) e^{2 z}}{z^{3}}

c a l c u l a t e R e s (f, 0)

Commented by prof Abdo imad last updated on 15/Jun/18

0 i s a t r i p l e p o l e o f f s o

R e s (f, 0) = {l i m}_{z \to 0} \frac{1}{(3 - 1)!} {z^{3} f (z)}^{(2)}

= {l i m}_{z \to 0} \frac{1}{2} {(1 - z^{2}) e^{2 z}}^{(2)} b u t

{{(1 - z^{2})}^{} e^{2 z}}^{(1)} = - 2 z e^{2 z} + 2 (1 - z^{2}) e^{2 z}

= (- 2 z + 2 - 2 z^{2}) e^{2 z} \Rightarrow

{(1 - z^{2}) e^{2 z}}^{(2)} = (- 4 z - 2) e^{2 z} + 2 (- 2 z + 2 - 2 z^{2}) e^{2 z}

= (- 4 z - 2 - 4 z + 4 - 4 z^{2}) e^{2 z}

= (- 4 z^{2} - 8 z + 2) e^{2 z} \Rightarrow

R e s (φ, 0) = \frac{1}{2} 2 = 1 .