Q1P - Mathematical Methods in Physical Sciences

Question

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

Step-by-Step Solution

Answer

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

Step 1: Definition of Power series.

The numbers that are presented in the form of $a+ib$where,$\mathit{a}\mathbf{,}\mathit{b}$ are real numbers and$\mathbf{\text{'}}\mathit{i}\mathbf{\text{'}}$  is an imaginary number called complex numbers.

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

Step 2: Substitute the value.

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

Substitute $\theta =-\frac{\pi }{4}$

$\begin{array}{c}{e}^{\frac{-i\pi }{4}}=\cos \left(\frac{-\pi }{4}\right)+i\sin \left(\frac{-\pi }{4}\right)\\ =\cos \frac{\pi }{4}-i\sin \frac{\pi }{4}\left\{\because \cos (-\theta )=\cos \theta \text{ and }\sin (-\theta )=-\sin \theta \right\}\\ =\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}\end{array}$

Step 3:Sketch the diagram.

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

 

Draw the diagram in the coordinates.

 

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