Q22P - Mathematical Methods in Physical Sciences

Question

Find each of the following in rectangular $\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$  form if  $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}\mathit{z}\mathbf{3}$ ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{z}\mathit{y}$ .

$\frac{\mathbf{1}}{\mathbf{z}\mathbf{-}\mathbf{1}}$

Step-by-Step Solution

Answer

The value for $\frac{1}{z-i}$  is given below

$$\begin{gathered} \frac{1}{z-i}=\frac{1+2i}{10} \\ \frac{1}{z-i}=\frac{x-\left(y-1\right)i}{x^2+{\left(y-1\right)}^2} \end{gathered}$$

 

Step 1: Given Information

The complex number is $\frac{1}{z-i}$.

Step 2: Definition of the Complex number.

Complex number z can be written in the form of the real and imaginary numbers as:

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$

Here z is the complex number, and a and b are real numbers

Step 3: Find the value.

The complex number is $\frac{1}{z-1}$ .

 

Find the values for 2-3i  .

z=2-3i 

$$\begin{gathered} \frac{1}{z-i}=\frac{1}{2-4i} \\ =\frac{1}{2-4i}\times \frac{2+4i}{2+4i} \\ =\frac{2+4i}{{\left(4-4i\right)}^2} \\ =\frac{1+2i}{10} \end{gathered}$$

 

Find the values for x+iy  .

$$\begin{gathered} z=x+iy \\ \frac{1}{z-1}=\frac{1}{\left(y-1\right)i} \\ =\frac{1}{x+\left(y-1\right)}\times \frac{x-\left(y-1\right)i}{x-\left(y-1\right)i} \\ =\frac{x-\left(y-1\right)i}{x^2+{\left(y-1\right)}^2} \end{gathered}$$

 

 

The value for $\frac{1}{z-1}$  is given below:

$$\begin{gathered} \frac{1}{z-1}=\frac{1+2i}{10} \\ \frac{1}{z-1}=\frac{x-\left(y-1\right)i}{x^2+{\left(y-1\right)}^2} \end{gathered}$$