91Ó°ÊÓ

Formula from Riemann surface hypothesis:

$w=R{e}^{i\mathrm{ϕ}}andz=r{e}^{i\mathrm{θ}}$

If$z=r{e}^{i\mathrm{θ}}withr>0$

$w=\ln r+i\mathrm{θ}$

It's one logarithm of z

Adding integer multiples of $2\mathrm{Ï€¾\pm }$.

$w=\ln r+i\left(\mathrm{θ}+2n\mathrm{π}\right),n∈Z$

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

Let's assume the following forms as follows:

$z=r{e}^{i\mathrm{θ}}$ ...... (2)

$w=R{e}^{i\mathrm{Ï•}}$ ...... (3)

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

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

$R{e}^{i\mathrm{ϕ}}=r^3{e}^{3i\mathrm{θ}}$

By equating each like terms, we get

$R=r^3$And$\mathrm{ϕ}=3\mathrm{θ}$ .

If r-constant for example, r = 1 and it keeps changing the value of $\mathrm{θ}$ then the value of $\mathrm{ϕ}$will change 3 times of $\mathrm{θ}$.

For example, if $\mathrm{θ}=\frac{\mathrm{π}}{2}$ then the image of it will be the.$\mathrm{ϕ}=\frac{3\mathrm{π}}{2}$

If r changes for example, r = 2 and keep the value of $\mathrm{θ}$ constant then the value of $\mathrm{ϕ}$ be .

If changes for example and keep changing the value of then the value of will changes 3 times, for example, if $\mathrm{θ}=\frac{\mathrm{π}}{2}$ then the image of will be the $\mathrm{ϕ}=\frac{3\mathrm{π}}{2}$.

Hence, $R=r^3$ and $\mathrm{ϕ}=3\mathrm{θ}$.