Question

Describe the Riemann surface for w = z³

Short answer

The required answer is

R = r³ and$\varphi =3\theta$

Step-by-step solution

Riemann surface hypothesis

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 $2\mathrm{\pi i}$.

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

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$\varphi =3\theta$ .

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 $\varphi =3\theta$.