C-e-z-1-cos-z-dz-C-z-1-

Question Number 115009 by arcana last updated on 22/Sep/20

\int_{C} \frac{e^{z}}{1 - \cos z} d z; C :∣ z ∣= 1

Answered by Olaf last updated on 24/Sep/20

\int_{C} \frac{co z + i \sin z}{1 - \cos z} d z

\int_{C} \frac{1 + i \sin z + (\cos z - 1)}{1 - \cos z} d z

\int_{C} \frac{d z}{1 - \cos z} + i \int_{C} \frac{\sin z}{1 - \cos z} - \int_{C} d z

\int_{C} \frac{2 d z}{\sin^{2} \frac{z}{2}} + i \int_{C} \frac{\sin z}{1 - \cos z} - \int_{C} d z

{[- 4 \cot \frac{z}{2} + i \ln (1 - \cos z) - z]}_{C}

Let z = e^{i θ}, θ \in [0; 2 π]

{[- 4 \cot \frac{e^{i θ}}{2} + i \ln (1 - \cos e^{i θ}) - e^{i θ}]}_{0}^{2 π}

= 0

Answered by Bird last updated on 24/Sep/20

i g i v e t h i s s o l u t i o n b u t n o t s u r e

I = \int_{C} \frac{e^{z}}{1 - c o s z} d z w e h a v e

1 - c o s z \sim \frac{z^{2}}{2} \Rightarrow \frac{e^{z}}{1 - c o s z} \sim \frac{2 e^{z}}{z^{2}}

s o o i s a d o u b l e p o l e f o r f (z) = \frac{e^{z}}{1 - c o s z}

I = 2 i π \times R e s (f, 0)

R e s (f, o) = {l i m}_{z \to 0} \frac{1}{(2 - 1)!} {z^{2} \frac{e^{z}}{1 - c o s z}}^{(1)}

= {l i m}_{z \to 0} {2 z^{2} e^{z}}^{(1)}

= {l i m}_{z \to 0} 2 {2 z e^{z} + 2 z^{2} e^{z}}

= 0

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