Q18P - Mathematical Methods in Physical Sciences

Question

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

$\sqrt{\mathbf{z}}$

Step-by-Step Solution

Answer

The real part $u\left(x,y\right)$ of the function is ${\left(x^2+y^2\right)}^{\frac{1}{4}}\cos \left(\frac{\theta }{2}\right)$, and the imaginary part $v\left(x,y\right)$of the function is, ${\left(x^2+y^2\right)}^{\frac{1}{4}}\sin \left(\frac{\theta }{2}\right)$, where $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$ .

 

Step 1: Definition of a complex number.

Complex numbersare 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 $\mathit{u}\left(x,y\right)$is the real part and $\mathit{v}\left(x,y\right)$ is the imaginary part.

 

Polar form of complex number is represented as: 

$\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{i}\mathit{y}\mathbf{ }\mathbf{=}\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$,where $\mathit{r}\mathbf{ }\mathbf{=}\mathbf{ }\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$, $\mathit{\theta }\mathbf{=}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\frac{y}{x}\right)$

Step 2: Solve complex number.

Given the function is $\sqrt{z}$ .

The complex number z can be written as:

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

 

Substitute the complex number and simplify.

$$\begin{gathered} \sqrt{z}={\left(x+iy\right)}^{\frac{1}{2}} \\ ={\left(r{e}^{i\theta }\right)}^{\frac{1}{2}} \end{gathered}$$

$$\begin{gathered} \sqrt{z}={\left(r{e}^{i\theta }\right)}^{\frac{1}{2}} \\ =r^{\frac{1}{2}}{e}^{\frac{i\theta }{2}} \\ =r^{\frac{1}{2}}\left(\cos \left(\frac{\theta }{2}\right)+i\sin \left(\frac{\theta }{2}\right)\right)----\left(1\right) \end{gathered}$$

where $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Step 3: Find real and imaginary parts.

Simplify the equationfurther.

 $$\begin{gathered} \sqrt{z}=r^{\frac{1}{2}}\left(\cos \left(\frac{\theta }{2}\right)+i\sin \left(\frac{\theta }{2}\right)\right) \\ ={\left[{\left(x^2+y^2\right)}^{\frac{1}{2}}\right]}^{\frac{1}{2}}\left(\cos \left(\frac{\theta }{2}\right)+i\sin \left(\frac{\theta }{2}\right)\right) \\ ={\left(x^2+y^2\right)}^{\frac{1}{4}}\left(\cos \left(\frac{\theta }{2}\right)+i\sin \left(\frac{\theta }{2}\right)\right) \\ ={\left(x^2+y^2\right)}^{\frac{1}{4}}\cos \left(\frac{\theta }{2}\right)+i{\left(x^2+y^2\right)}^{\frac{1}{4}}\sin \left(\frac{\theta }{2}\right) \end{gathered}$$

 

Hence, the real part is ${\left(x^2+y^2\right)}^{\frac{1}{4}}\cos \left(\frac{\theta }{2}\right)$ and imaginary part is $i{\left(x^2+y^2\right)}^{\frac{1}{4}}\sin \left(\frac{\theta }{2}\right)$, where $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$ .