Chapter 14: Q25P (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.
$\frac{e^{2 z} - 1}{z^{2}} at z = 0$

Short Answer

Expert verified

Hence, the residue of the function at $z_{0} = 0$ is 2.

Step by step solution

Given information

The given function is: $f (z) = \frac{e^{2 z} - 1}{z^{2}}$ .

Residue Theorem

If $z_{0}$ is an isolated singular point of $f (z)$ . Then the integration of the function within any closed curve C is given by:

$\oint_{c} f (z) d z = b_{1} \cdot 2 π i$

Here, $b_{1}$ is the residue.

Step 3:Find the Residue

Since, $z = 0$ is a pole of higher order of 2.

So, the residue of this type of function is given by:

$\underset{z = z_{0}}{R e s} f (z) = \frac{1}{(n - 1)!} \lim_{z \to z_{0}} \{\frac{d^{n - 1}}{d z^{n - 1}} (f (z) \cdot {(z - z_{0})}^{n})\}$

According to the question, we have: $z_{0} = 0 and n = 2$

Now, the residue at $z_{0} = 0$ will be:

$\begin{array}{rcl} \underset{z = 0}{R e s} f (z) & = & \frac{1}{(2 - 1)!} \lim_{z \to 0} \{\frac{d^{2 - 1}}{d z^{2 - 1}} ((\frac{e^{2 z} - 1}{z^{2}}) \cdot {(z - 0)}^{2})\} \\ = & \frac{1}{(1)!} \lim_{z \to 0} \{\frac{d}{d z} (e^{2 z} - 1)\} \\ = & \lim_{z \to 0} \{2 e^{2 z}\} \\ = & 2 \end{array}$

Hence, the residue of the function at $z_{0} = 0$ is 2.

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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.
$\frac{e^{2 z} - 1}{z^{2}} at z = 0$

Short Answer

Step by step solution

Given information

Residue Theorem

Step 3:Find the Residue

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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.e2z-1z2atz=0

Short Answer

Step by step solution

Given information

Residue Theorem

Step 3:Find the Residue

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Dynamics

Electricity

Rotational Dynamics

Electricity and Magnetism

Modern Physics

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.
$\frac{e^{2 z} - 1}{z^{2}} at z = 0$