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Complex Analysis : Residue Theory

Study concepts, example questions & explanations for Complex Analysis

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Example Questions

Example Question #51 : Complex Analysis

Find the residue at z = 0 for the function

$f(z) = \frac{4}{z+z^2}$ .

Possible Answers:

$\frac{1}{4}$

$\pi$

$\frac{1}{2}$

Correct answer:

Explanation:

Observe,

$f(z) \newline = \frac{4}{z + z^2} \newline =\frac{4}{z} * \frac{1}{z+1} \newline = \frac{4}{z} \displaystyle\sum_{k=0}^{\infty} (-1)^{k} z^{k} \newline = 4\displaystyle\sum_{k=0}^{\infty} (-1)^{k} z^{k -1} \newline = 4(\frac{1}{z} -1 + z - z^2 +z^3 + \cdots)$

The coefficient of $\frac{1}{z}$ is 4*1=4 .

Thus,

$\text{Res}_{z=0} [f(z)] = \text{Res}_{z=0} [\frac{4}{z+z^2}] = 4$ .

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Example Question #52 : Complex Analysis

Find the residue at z=0 of the function

$f(z) = \frac{z^2-sin(z)}{z^2}$ .

Possible Answers:

$-\frac{1}{2}$

$\frac{7}{9}$

Correct answer:

Explanation:

Observe,

$f(z) \newline = \frac{z-sin(z)}{z} \newline = 1 - \frac{sin(z)}{z^2} \newline = 1 - \displaystyle\sum_{k=0}^{\infty} \frac{-1^{k}}{(2n+1)!} z^{2n-1}$ .

The coefficient of $\frac{1}{z}$ is $\frac{(-1)^0}{(2*0+1)!}= 1$ .

Thus,

$\text{Res}_{z=0} [f(z)] = \text{Res}_{z=0} [\frac{z^2 - sin(z)}{z^2}] = -1$ .

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Complex Analysis : Residue Theory

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #51 : Complex Analysis

Example Question #52 : Complex Analysis

All Complex Analysis Resources

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