Q2P - 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.${\mathit{e}}^{\mathbf{i}\mathbf{\pi }\mathbf{/}\mathbf{2}}$

 

Step-by-Step Solution

Answer

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

Step 1: Definition of Power series.

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

Example: .

 $3+2i$

Step 2: Substitute the value.

 

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

Substitute $\theta =\frac{\pi }{2}$

$\begin{array}{c}{e}^{\frac{i\pi }{2}}=\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\\ =0+i\left(1\right)\\ =i\end{array}$

Step 3: Sketch the diagram.

Write the given expression form of $r{e}^{i\theta }$for that .$r=1\text{ and }\theta =\frac{\pi }{2}$

 

Draw the diagram in the coordinates.

Therefore, the general form of the given complex number is.$z=i$