f-z-cosz-1-sin-z-2-find-residus-of-f-

Question Number 148213 by mathmax by abdo last updated on 26/Jul/21

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

find residus of f

Answered by puissant last updated on 26/Jul/21

\sin (z^{2}) ∽ z^{2} - \frac{z^{6}}{6} \Rightarrow - \sin (z^{2}) ∽ - z^{2} + \frac{z^{6}}{6}

\Rightarrow 1 - \sin (z^{2}) ∽ 1 - z^{2} + \frac{z^{6}}{6} ∽ z^{7} (\frac{1}{z^{7}} - \frac{1}{z^{5}} + \frac{1}{6 z})

\Rightarrow f (z) ∽ \frac{\cos (z)}{z^{7} (\frac{1}{z^{7}} - \frac{1}{z^{5}} + \frac{1}{6 z})}

or \cos (z) = \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{(2 n)!} z^{2 n}

∽ 1 - \frac{z^{2}}{2!} + \frac{z^{4}}{4!} - \frac{z^{6}}{6!} + \frac{z^{8}}{8!} \dots

\Rightarrow f (z) ∽ \frac{1}{z^{7}} (1 - \frac{z^{2}}{2} + \frac{z^{4}}{4!} - \frac{z^{6}}{6!} + \frac{z^{8}}{8!}) \times \frac{1}{(\frac{1}{z^{7}} - \frac{1}{z^{5}} + \frac{1}{6 z})}

\Rightarrow f (z) ∽ (\frac{1}{z^{7}} - \frac{1}{2! z^{5}} + \frac{1}{4! z^{3}} - \frac{1}{6! z}) \times \frac{1}{(\frac{1}{z^{7}} - \frac{1}{z^{5}} + \frac{1}{6 z})}

Res (f) = - \frac{1}{6!} \times 6 = - \frac{1}{120} \dots

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