If-C-is-circle-z-1-Then-the-value-of-C-cos-z-sin-z-dz-

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Question Number 124 by novrya last updated on 25/Jan/15

$$IfCiscircle\mid z\mid =1.Thenthevalueof$$$$\underset{C}{\int }\frac{cosz}{sinz}dz=\dots .$$

Commented by 123456 last updated on 13/Dec/14

$$\mathrm{o}\mathrm{teorema}\mathrm{dos}\mathrm{residuos}\mathrm{seria}\mathrm{bem}\mathrm{util}\mathrm{aqui}.$$$$\mathrm{note}\mathrm{que}\mathrm{no}\mathrm{interior}\mathrm{do}\mathrm{contorno}\mathrm{C}\mathrm{so}\mathrm{a}\mathrm{um}\mathrm{polo}\mathrm{simples}(\mathrm{z}=0)$$$$\mathrm{o}\mathrm{resto}\mathrm{fica}\mathrm{como}\mathrm{exercicio}\dots$$

Answered by 123456 last updated on 14/Dec/14

$$\mathrm{we}\mathrm{have}\mathrm{that}\mathrm{C}\mathrm{have}\mathrm{only}\mathrm{one}\mathrm{simple}\mathrm{pole}\mathrm{at}\mathrm{z}=0\mathrm{then}$$$$\mathrm{res}(\mathrm{f},0)=\underset{x\to 0}{\lim }\frac{x\cos x}{\sin x}=1$$$$\mathrm{then}\mathrm{by}\mathrm{residue}\mathrm{theorem}$$$${\int }_C\frac{\cos z}{\sin z}dz=2\pi i\mathrm{res}(\mathrm{f},0)=2\pi i$$

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