Chapter 2: Q1P (page 63)

Express the following complex numbers in the $x + i y$ form. Try to visualize each complex number, using sketches as in the examples if necessary $e^{- i π / 4}$ .

Short Answer

Expert verified

The complex number $e^{- i π / 4}$ in the $x + i y$ form is $\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}$ and the sketch is given below.

Step by step solution

Definition of Power series.

The numbers that are presented in the form of $a + i b$ where, $a, b$ arereal numbers and $' i'$ is an imaginary number called complex numbers.

Example: . $3 + 2 i$ $3 + 2 i$

Substitute the value.

It is known that $e^{i θ} = \cos θ + i \sin θ$

Substitute $θ = - \frac{π}{4}$

$\begin{matrix} e^{\frac{- i π}{4}} = \cos (\frac{- π}{4}) + i \sin (\frac{- π}{4}) \\ = \cos \frac{π}{4} - i \sin \frac{π}{4} {∵ \cos (- θ) = \cos θ and \sin (- θ) = - \sin θ} \\ = \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \end{matrix}$

Step 3:Sketch the diagram.

Write the given expression form of $r e^{i θ}$ for that.

Draw the diagram in the coordinates.

Therefore, the general form of the given complex number is $\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}$

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.

Express the following complex numbers in the $x + i y$ form. Try to visualize each complex number, using sketches as in the examples if necessary $e^{- i π / 4}$ .

Short Answer

Step by step solution

Definition of Power series.

Substitute the value.

Step 3:Sketch the diagram.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Space Physics

Mechanics and Materials

Medical Physics

Linear Momentum

Electrostatics

Study anywhere. Anytime. Across all devices.

Express the following complex numbers in thex+iyform. Try to visualize each complex number, using sketches as in the examples if necessarye−iπ/4.

Short Answer

Step by step solution

Definition of Power series.

Substitute the value.

Step 3:Sketch the diagram.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Space Physics

Mechanics and Materials

Medical Physics

Linear Momentum

Electrostatics

Study anywhere. Anytime. Across all devices.

Express the following complex numbers in the $x + i y$ form. Try to visualize each complex number, using sketches as in the examples if necessary $e^{- i π / 4}$ .