Chapter 14: Q22P (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.
$\frac{e^{2 z}}{1 + e^{z}} at z = i π$

Short Answer

Expert verified

Hence, the residue of the function at $z = i π$ is -1 .

Step by step solution

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.

Find the Residue

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

As we know, the residue of the function is given by:

$R (z = z_{0}) = \frac{g (z_{0})}{h' (z_{0})}$ , for $f (z) = \frac{g (z)}{h (z)}$

According to the question, we have:

$g (z) = e^{2 z} h (z) = 1 + e^{z} \Rightarrow h' (z) = e^{z}$

Now, the residue for $z_{0} = i π$ will be:

$\begin{array}{rcl} R (i π) & = & \frac{g (i π)}{h' (i π)} \\ = & \frac{e^{2 i π}}{e^{i π}} \\ = & e^{i π} \\ = & \cos π + i \sin π . . . . . . . . . . . . \{e^{i θ} = \cos θ + i \sin θ\} \end{array}$

Simplication

Simplifying further, we get:

$\begin{array}{rcl} R (i π) & = & \cos π + i \sin π \\ = & - 1 + i \cdot 0 \\ = & - 1 + 0 \\ = & - 1 \end{array}$

Hence, the residue of the function at $z = i π$ is -1.

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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.
$\frac{e^{2 z}}{1 + e^{z}} at z = i π$

Short Answer

Step by step solution

Residue Theorem

Find the Residue

Simplication

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.e2z1+ezatz=iπ

Short Answer

Step by step solution

Residue Theorem

Find the Residue

Simplication

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Magnetism and Electromagnetic Induction

Atoms and Radioactivity

Electromagnetism

Linear Momentum

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.
$\frac{e^{2 z}}{1 + e^{z}} at z = i π$