Question
Let $f(z)=\frac{\left(z^2+1\right)^2}{\left(z^2+2 z+2\right)^3}$. Evaluate $\frac{1}{2 \pi i} \oint_C \frac{f^{\prime}(z)}{f(z)} d z$ where $C$ is the circle $|z|=4$.
Step 1
The poles occur when the denominator of $f(z)$ is equal to 0. So, we solve the equation $z^2+2z+2=0$ to find the poles. Show more…
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Use (7.8) to evaluate $$\oint_{C} \frac{f^{\prime}(z)}{f(z)} d z, \quad \text { where } \quad f(z)=\frac{z^{3}(z+1)^{2} \sin z}{\left(z^{2}+1\right)^{2}(z-3)}$$ around the circle $|z|=2 ;$ around $|z|=\frac{1}{2}$
Functions of a Complex Variable
Evaluation of Definite Integrals by Use of the Residue Theorem
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