Q16 MP - Mathematical Methods in Physical Sciences

Question

Describe the set of points $\mathit{z}$for which $\mathbf{Re}\left(e^{i\pi /2}z\right)\mathbf{>}\mathbf{2}$.

Step-by-Step Solution

Answer

The set of points are $-\infty <y<-2$.

Step 1: Given Information

The given relation is $Re\left(e^{i\mathrm{\pi }/2}\right)>2$.

Step 2: Definition of Complex Numbers

The numbers that are presented in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$where, $\mathbf{\text{'}}\mathit{x}\mathbf{\text{'}}$ is real numbers and $\mathbf{\text{'}}\mathit{i}\mathit{y}\mathbf{\text{'}}$ is an imaginary number, those numbers are referred to as called Complex numbers.

Step 3: Find the ratio of the series.

The given relation is, $Re\left(e^{i\mathrm{\pi }/2}z\right)>2$.

 

Write $e^{i\mathrm{\pi }/2}$in polar form.

$$\begin{gathered} e^{i\mathrm{\pi }/2}=\cos \left(\mathrm{\pi }/2\right)+i\sin \left(\mathrm{\pi }/2\right) \\ =i \end{gathered}$$

 

Let $z=x=yi$

Find $\mathrm{Re}\left(zi\right)>2$, the real part of the complex number.

$$\begin{gathered} Re\left(zi\right)=Re\left[i\left(x+yi\right)\right] \\ =Re\left(xi-y\right) \\ =-y \\ -y>2 \end{gathered}$$

Or  $y<-2$ .

 

So, the set of points are $-\infty <y<-2$.