Q9P - Mathematical Methods in Physical Sciences

Question

Describe the Riemann surface for  w = z³

Step-by-Step Solution

Answer

The required answer is  

R = r³ and $\varphi =3\theta$

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  $2\mathrm{\pi i}$.

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

Step 2: Use Riemann surface hypothesis

Function is given as .      w = z³                                                                              ...... (1)

Let's assume the following forms as follows:

      $z=r{e}^{i\theta }$                                                                                                                  ...... (2)

          $w=R{e}^{i\varphi }$                                                                                                           ...... (3)

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

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

 $R{e}^{i\varphi }=r^3{e}^{3i\theta }$

By equating each like terms, we get 

$R=r^3$ And $\varphi =3\theta$ .

Step 3: Put the different values of R

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 $\theta =\frac{\pi }{2}$  then the image of it will be the .$\varphi =\frac{3\pi }{2}$

If   r changes for example, r = 2  and keep the value of  $\theta$ constant then the value of  $\varphi$ be  .

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

Hence,  $R=r^3$ and $\varphi =3\theta$.