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Question Number 147406 by tabata last updated on 20/Jul/21

$$find{\int }_C\frac{z+2}{sin\left(\frac{z}{2}\right)}dz,\mid z\mid =3\pi$$

Answered by Olaf_Thorendsen last updated on 20/Jul/21

$$\mathrm{\Omega }={\int }_{\mid z\mid =3\pi }\frac{z+2}{\sin \left(\frac{z}{2}\right)}dz$$$$\mathrm{\Omega }={\int }_{\mid z\mid =3\pi }\frac{z+2}{z}\times \frac{z}{\sin \left(\frac{z}{2}\right)}dz$$$$\mathrm{\Omega }=2i\pi \underset{z\to 0}{\lim }\left(\frac{z(z+2)}{\sin \left(\frac{z}{2}\right)}\right)$$$$\mathrm{\Omega }=2i\pi \underset{z\to 0}{\lim }\left(\frac{2z+2}{\frac{1}{2}\cos \left(\frac{z}{2}\right)}\right)$$$$\mathrm{\Omega }=2i\pi \times 4=8i\pi$$

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