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

$$\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{cosz}}{1-\sin \left({\mathrm{z}}^2\right)}$$$$\mathrm{find}\mathrm{residus}\mathrm{of}\mathrm{f}$$

Answered by puissant last updated on 26/Jul/21

$$\sin \left({\mathrm{z}}^2\right)\backsim {\mathrm{z}}^2-\frac{{\mathrm{z}}^6}{6}\Rightarrow -\sin \left({\mathrm{z}}^2\right)\backsim -{\mathrm{z}}^2+\frac{{\mathrm{z}}^6}{6}$$$$\Rightarrow 1-\sin \left({\mathrm{z}}^2\right)\backsim 1-{\mathrm{z}}^2+\frac{{\mathrm{z}}^6}{6}\backsim {\mathrm{z}}^7(\frac{1}{{\mathrm{z}}^7}-\frac{1}{{\mathrm{z}}^5}+\frac{1}{6\mathrm{z}})$$$$\Rightarrow \mathrm{f}\left(\mathrm{z}\right)\backsim \frac{\cos \left(\mathrm{z}\right)}{{\mathrm{z}}^7(\frac{1}{{\mathrm{z}}^7}-\frac{1}{{\mathrm{z}}^5}+\frac{1}{6\mathrm{z}})}$$$$\mathrm{or}\cos \left(\mathrm{z}\right)=\underset{\mathrm{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{\mathrm{n}}}{\left(2\mathrm{n}\right)!}{\mathrm{z}}^{2\mathrm{n}}$$$$\backsim 1-\frac{{\mathrm{z}}^2}{2!}+\frac{{\mathrm{z}}^4}{4!}-\frac{{\mathrm{z}}^6}{6!}+\frac{{\mathrm{z}}^8}{8!}...$$$$\Rightarrow \mathrm{f}\left(\mathrm{z}\right)\backsim \frac{1}{{\mathrm{z}}^7}(1-\frac{{\mathrm{z}}^2}{2}+\frac{{\mathrm{z}}^4}{4!}-\frac{{\mathrm{z}}^6}{6!}+\frac{{\mathrm{z}}^8}{8!})\times \frac{1}{(\frac{1}{{\mathrm{z}}^7}-\frac{1}{{\mathrm{z}}^5}+\frac{1}{6\mathrm{z}})}$$$$\Rightarrow \mathrm{f}\left(\mathrm{z}\right)\backsim (\frac{1}{{\mathrm{z}}^7}-\frac{1}{2!{\mathrm{z}}^5}+\frac{1}{4!{\mathrm{z}}^3}-\frac{1}{6!\mathrm{z}})\times \frac{1}{(\frac{1}{{\mathrm{z}}^7}-\frac{1}{{\mathrm{z}}^5}+\frac{1}{6\mathrm{z}})}$$$$\mathrm{Res}\left(\mathrm{f}\right)=-\frac{1}{6!}\times 6=-\frac{1}{120}...$$

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