Describe the Riemann surface for .$\mathit{w}\mathbf{=}\sqrt{\mathbf{z}}$

Formula from Riemann surface hypothesis:

$w=R{e}^{i\varphi }andz=r{e}^{i\theta }$

If $z=r{e}^{i\theta }withr>0$

$w=\ln r+i\theta$

It's one logarithm of z.

Adding integer multiples of$2\mathrm{\pi I}$ .

$w=\ln r+i\left(\theta +2n\pi \right),n\in Z$

Function is given as $w=\sqrt{z}$

Let's assume the following forms as follows:

$z=r{e}^{i\theta }$ ...... (1)

$w=R{e}^{i\varphi }$ ...... (2)

To find the surface w first change r how that impact on R .

$w=\sqrt{z}$ ...... (3)

Put (1) and (2) in equation (3) as follows:

$R{e}^{i\varphi }=\sqrt{r{e}^{i\theta }}$

$R{e}^{i\varphi }=r^{\frac{1}{2}}{e}^{\frac{1}{2}i\theta }$

By equating each like terms, we get

$R=r^{\frac{1}{2}}$And$\varphi =\frac{\theta }{2}$ or $2\varphi =\theta$

So if r = constant for example, r = 1 and it keeps changing the value of $\theta$ then the value of $\varphi$ will change 3 times of$\theta$. For example, if $\theta =\pi$ then the image of it will be the$\varphi =\frac{\pi }{2}$.

If r changes for example, r = 2 and keep the value of $\theta$ constant then the value of R be $r^{\frac{1}{2}}=1.41$.

If r changes for example r = 2 and keep changing the value of $\theta$ then the value of $\varphi$ will changes 3 times, for example, if $\theta =\pi$ then the image of w will be the$\varphi =\frac{\pi }{2}$ .

Hence, $R=r^{\frac{1}{2}}$and$2\varphi =\theta$ .