Chapter 14: Q9P (page 667)

Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$\frac{1}{z}$

Short Answer

Expert verified

The real part $u (x, y)$ of the function is $\frac{x}{x^{2} + y^{2}}$ and the imaginary part $v (x, y)$ of the function is $\frac{y}{x^{2} + y^{2}}$ .

Step by step solution

Definition of complex number

Complex number is represented by, where $a + i b$ is the a real number and bis the imaginary number.

Solve complex number

Given the function is $\frac{1}{z}$ .

The complex number z can be written as: $\frac{1}{x + i y}$

$z = x + i y$ , where x is a real part and y is an imaginary part.

The function $\frac{1}{z}$ can be written as

Multiply with complex conjugate of $z = x + i y$ into denominator and numerator to separate real and imaginary parts.

$\frac{1}{z} = \frac{(x - i y)}{(x + i y) (x - i y)} = \frac{(x - i y)}{x^{2} - i^{2} y^{2}} = \frac{(x + i y)}{x^{2} + y^{2}}$

Find real and imaginary parts

Simplify the expression future

\frac{(x + i y)}{x^{2} + y^{2}} = \frac{x}{x^{2} + y^{2}} + i \frac{y}{x^{2} + y^{2}}

Hence the real part is $\frac{x}{x^{2} + y^{2}}$ and imaginary part is $\frac{y}{x^{2} + y^{2}}$ .

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 real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$\frac{1}{z}$

Short Answer

Step by step solution

Definition of complex number

Solve complex number

Find real and imaginary parts

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Translational Dynamics

Electrostatics

Wave Optics

Fields in Physics

Radiation

Study anywhere. Anytime. Across all devices.

Find the real and imaginary parts u(x,y)and v(x,y)of the following functions.1z

Short Answer

Step by step solution

Definition of complex number

Solve complex number

Find real and imaginary parts

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Translational Dynamics

Electrostatics

Wave Optics

Fields in Physics

Radiation

Study anywhere. Anytime. Across all devices.

Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$\frac{1}{z}$