Q10P - Mathematical Methods in Physical Sciences

Question

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

Step-by-Step Solution

Answer

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

Step 1: Riemann surface hypothesis

Formula from Riemann surface hypothesis:

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

If  $z=r{e}^{i\theta } with r>0$ 

 $w=\ln r+i\theta$

It's one logarithm of z.

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

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

Step 2: Use Riemann surface hypothesis

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  $2\varphi =\theta$

Step 3: Put different values of R

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 $2\varphi =\theta$ .