Chapter 14: Q32P (page 687)

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{iz}}{{(z^{2} + 4)}^{2}} at z = 2 i$

Short Answer

Expert verified

The residue of the function at $z = 2 i$ is $R (2 i) = \frac{3 i}{32} e^{- 2}$

Step by step solution

Determine the formula for the higher order poles

Residue of a function at higher order poles is given as follows:

$R[f(z)]= \frac{1}{(n - 1)!} \lim_{z \to z_{k}} \frac{d^{n - 1}}{{dz}^{n - 1}} {(z - z_{k})}^{n} f (z)$

Determine the residue of the higher order pole:

Consider the given function as:

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

Resolve the function as:

$\begin{array}{rcl} f (z) & = & \frac{e^{i z}}{{(z^{2} + 4)}^{2}} \\ = & \frac{e^{i z}}{{(z^{2} + 2 i)}^{2} {(z - 2 i)}^{2}} \end{array}$

At z = 2i

The function has a pole of order n = 2 at z = 2i and the residue of higher order pole is given by:

$R (z_{0}) = \frac{1}{(n - 1)!} \lim_{z \to z_{0}} \frac{d^{n - 1}}{d z^{n - 1}} (z - z_{0}) f (z)$ ……. (1)

Determine the residue of the higher order function as:

Substitute the values in equation (1) and solve as:

$\begin{array}{rcl} R (2 i) & = & \frac{1}{(n - 1)!} \lim_{z \to 0} \frac{d}{d z} (\frac{e^{i z}}{{(z + 2 i)}^{2}}) \\ = & {[\frac{i e^{i z} {(z + 2 i)}^{2} - 2 e^{i z} (z + 2 i)}{{(z + 2 i)}^{4}}]}_{z = 2 i} \\ = & {[\frac{i e^{i z} (z + 2 i) - 2 e^{i z}}{{(z + 2 i)}^{3}}]}_{z = 2 i} \end{array}$

Solve further as:

$\begin{array}{rcl} R (2 i) & = & {[\frac{i e^{i z} (z + 2 i) - 2 e^{i z}}{{(z + 2 i)}^{3}}]}_{z = 2 i} \\ = & [\frac{i e^{i z} (4 i) - e^{- 2} \times 2}{{(4 i)}^{3}}] \\ = & \frac{3 i}{32} e^{- 2} \end{array}$

Therefore, the residue of a function at z = 2i is $\begin{array}{rcl} R (2 i) & = & \frac{3 i}{32} e^{- 2} \end{array}$ .

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Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{iz}}{{(z^{2} + 4)}^{2}} at z = 2 i$

Short Answer

Step by step solution

Determine the formula for the higher order poles

Determine the residue of the higher order pole:

Determine the residue of the higher order function as:

One App. One Place for Learning.

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Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.eiz(z2+4)2atz=2i

Short Answer

Step by step solution

Determine the formula for the higher order poles

Determine the residue of the higher order pole:

Determine the residue of the higher order function as:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Waves Physics

Particle Model of Matter

Measurements

Conservation of Energy and Momentum

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{iz}}{{(z^{2} + 4)}^{2}} at z = 2 i$