Chapter 14: Q24P (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.
$\frac{1 - c o s 2 z}{z^{3}} at z = 0$

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: $f (z) = \frac{1 - \cos 2 z}{z^{3}}$ .

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 3.

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: $z_{0} = 0 and n = 3$

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

$\begin{array}{rcl} \underset{z = 0}{R e s} f (z) & = & \frac{1}{(3 - 1)!} \lim_{z \to 0} \{\frac{d^{3 - 1}}{d z^{3 - 1}} ((\frac{1 - \cos 2 z}{z^{3}}) \cdot {(z - 0)}^{3})\} \\ = & \frac{1}{(2)!} \lim_{z \to 0} \{\frac{d^{2}}{d z^{2}} (1 - \cos 2 z)\} \\ = & \frac{1}{(2)!} \lim_{z \to 0} \{4 \cos 2 z\} \\ = & \frac{1}{2} \cdot 4 = 2 \end{array}$

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.
$\frac{1 - c o s 2 z}{z^{3}} at z = 0$

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

Nuclear Physics

Further Mechanics and Thermal Physics

Particle Model of Matter

Electric Charge Field and Potential

Kinematics Physics

Waves 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.1-cos2zz3atz=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

Nuclear Physics

Further Mechanics and Thermal Physics

Particle Model of Matter

Electric Charge Field and Potential

Kinematics Physics

Waves 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.
$\frac{1 - c o s 2 z}{z^{3}} at z = 0$