Q5P - Mathematical Methods in Physical Sciences

Question

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

$\mathbf{Re}\mathit{z}$

Step-by-Step Solution

Answer

The real part $u\left(x,y\right)$ of the function is x , and the imaginary part $v\left(x,y\right)$of the function is 0.

Step 1: Definition of complex numbers

Complex numbers are expressed in the form of $\mathit{z}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{i}\mathit{y}$ , where x,y are real numbers, and i is an imaginary number.

 

Similarly, the function of z is represented as follows:

$\mathit{f}\left(z\right)\mathbf{=}\mathit{f}\left(x+iy\right)\mathbf{=}\mathit{u}\left(x,y\right)\mathbf{+}\mathit{i}\mathit{v}\left(x,y\right)$

, where $\mathit{u}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$ is the real part and $\mathit{v}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$ is the imaginary part.

Step 3: Substitute the value in the given function.

 

The given function is $Rez$ .

 

Substitute $z = x + iy$ in $Rez$as follows:

 $$\begin{gathered} Rez=Re\left(x+iy\right) \\ =x \\ =x+0i \end{gathered}$$

 

The above equation is in the form of $f\left(z\right)=f\left(x+iy\right)=u\left(x,y\right)+iv\left(x,y\right)$ , such that$u\left(x,y\right)=x$and $v\left(x,y\right)=0$ .

 

Hence, the real part of the given function is $u\left(x,y\right)=x$ , and the imaginary part is $v\left(x,y\right)=0$ .