evaluate-C-z-2-sin-z-2z-cos-z-pi-dz-C-2-2-sin-z-2z-cos-z-pi-dz-where-C-is-the-circle-z-3pi-4-pi-4-

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

evaluate

\frac{\oint_{C} \frac{z}{2 \cdot \sin (z) - 2 z \cdot \cos (z) - π} d z}{\oint_{C} \frac{2}{2 \cdot \sin (z) - 2 z \cdot \cos (z) - π} d z}

where C is the circle ∣ z - \frac{3 π}{4} ∣= \frac{π}{4} .

Answered by MrGaster last updated on 24/Dec/24

\oint_{C} \frac{z}{2 \cdot \sin (z) - 2 z \cdot \cos (z) - π} d z

\oint_{C} \frac{2}{2 \cdot \sin (z) - 2 z \cdot \cos (z) - π} d z

= \frac{\oint_{C} z d z}{\oint_{C} 1 d z}

\underset{= \frac{{C a u c h y}^{,} s i n e g r a l t h e o r e m}{2 π i \cdot Z}}{\underset{⏟}{\oint_{C} z d z = 0}}

= \frac{0}{2 π i \cdot Z}

= 0

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