Q9P - Mathematical Methods in Physical Sciences

Question

Find the real and imaginary parts $\mathit{u}\left(x,y\right)$and $\mathbf{v}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$of the following functions.

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

Step-by-Step Solution

Answer

The real part $u\left(x,y\right)$of the function is $\frac{x}{x^2+y^2}$and the imaginary part$v\left(x,y\right)$of the function is$\frac{y}{x^2+y^2}$ .

 

Step 1: Definition of complex number

Complex number is represented by , where $\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$  is the a real number and b is the imaginary number.

Step 2: Solve complex number

Given the function is $\frac{1}{z}$.

 

The complex number z can be written as:$\frac{1}{x+iy}$

 $z=x+iy$, where x is a real part and y is an imaginary part.

 

The function $\frac{1}{z}$can be written as

 

Multiply with complex conjugate of $z=x+iy$into denominator and numerator to separate real and imaginary parts.

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

Step 3: Find real and imaginary parts

 

Simplify the expression future

$\frac{\left(x+iy\right)}{x^2+y^2}=\frac{x}{x^2+y^2}+i\frac{y}{x^2+y^2}$

Hence the real part is $\frac{x}{x^2+y^2}$and imaginary part is $\frac{y}{x^2+y^2}$.