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Question Number 214340 by issac last updated on 06/Dec/24

$$\mathrm{evaluate}$$$$\frac{{\oint }_C\frac{z}{2\cdot \sin \left(z\right)-2z\cdot \cos \left(z\right)-\pi }\mathrm{d}z}{{\oint }_C\frac{2}{2\cdot \sin \left(z\right)-2z\cdot \cos \left(z\right)-\pi }\mathrm{d}z}$$$$\mathrm{where}C\mathrm{is}\mathrm{the}\mathrm{circle}\mid z-\frac{3\pi }{4}\mid =\frac{\pi }{4}.$$

Answered by MrGaster last updated on 24/Dec/24

$${\oint }_C\frac{z}{\cancel{2\cdot \sin \left(z\right)-2z\cdot \cos \left(z\right)-\pi }}dz$$$${\oint }_C\frac{2}{\cancel{2\cdot \sin \left(z\right)-2z\cdot \cos \left(z\right)-\pi }}dz$$$$=\frac{{\oint }_{C}zdz}{{\oint }_{C}1dz}$$$$\underset{=\frac{{Cauchy}^{,}sinegraltheorem}{2\pi i\cdot \mathbb{Z}}}{\underbrace{{\oint }_{C}zdz=0}}$$$$=\frac{0}{2\pi i\cdot \mathbb{Z}}$$$$=0$$

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