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Question Number 130369 by gowsalya last updated on 24/Jan/21

Answered by Olaf last updated on 24/Jan/21

$$\mathrm{\Omega }={\int }_{\mid z\mid =1}xdz$$$$\mathrm{\Omega }={\int }_0^{2\pi }\cos \theta d(\cos \theta +i\sin \theta )$$$$\mathrm{\Omega }={\int }_0^{2\pi }\cos \theta (-\sin \theta +i\cos \theta )d\theta$$$$\mathrm{\Omega }={\int }_0^{2\pi }(-\frac{1}{2}\sin 2\theta +i\frac{1+\cos 2\theta }{2})d\theta$$$$\mathrm{\Omega }={[\frac{1}{4}\cos 2\theta +i\frac{\theta +\frac{1}{2}\sin 2\theta }{2}]}_0^{2\pi }$$$$\mathrm{\Omega }=i\pi$$

Commented by gowsalya last updated on 25/Jan/21

thank u

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