Answer to Question #91339 in Complex Analysis for Ra

Question

Describe the geometric, polar and exponential representations of:
(\bar -5i+2)⁻¹

Answer 1

Let's find a conjugate to -5i+2

$\overline{-5i+2}=2+5i$

let's find the rectangular representation of this number:

$z=( 2+5i )^{-1}= \frac{1}{2+5i}= \frac{2-5i}{(2+5i)(2-5i)}= \frac{2-5i}{29}$

Real part

$Re(z)=\frac{2}{29}$

Imaginary part

$Im(z)=-\frac{5}{29}$

Modulus of z

$|z|=\sqrt{\frac{4}{29^2}+\frac{25}{29^2}}=\frac{\sqrt{29}}{29}$

Re(z)>0 and Im(z)<0 hence

$arg(z)=\theta=2\pi-arctan \frac{|Im(z)|}{Re(z)} ​ =2\pi-arctan \frac{\frac{5}{29}}{\frac{2}{29}}= 2\pi-arctan\frac{5}{2}$

Therefore the polar representation of z will be

$z=\frac{\sqrt{29}}{29}\left(cos\left(2\pi-arctan\frac{5}{2}\right )+isin\left(2\pi-arctan\frac{5}{2}\right )\right)$

Exponential representation of z

$z=\frac{\sqrt{29}}{29}e^{i\left(2\pi-arctan\frac{5}{2}\right )}$

Geometric representation