find-C-z-2-sin-z-2-dz-z-3pi-

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

f i n d \int_{C} \frac{z + 2}{s i n (\frac{z}{2})} d z, ∣ z ∣= 3 π

Answered by Olaf_Thorendsen last updated on 20/Jul/21

Ω = \int_{∣ z ∣= 3 π} \frac{z + 2}{\sin (\frac{z}{2})} d z

Ω = \int_{∣ z ∣= 3 π} \frac{z + 2}{z} \times \frac{z}{\sin (\frac{z}{2})} d z

Ω = 2 i π \lim_{z \to 0} (\frac{z (z + 2)}{\sin (\frac{z}{2})})

Ω = 2 i π \lim_{z \to 0} (\frac{2 z + 2}{\frac{1}{2} \cos (\frac{z}{2})})

Ω = 2 i π \times 4 = 8 i π