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$2\mathrm{Ï€\pm õ}$ .

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

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

Let's assume the following forms as follows:

$z=r{e}^{i\mathrm{θ}}$ ...... (1)

$w=R{e}^{i\mathrm{Ï•}}$ ...... (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:

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

$R{e}^{i\mathrm{ϕ}}=r^{\frac{1}{2}}{e}^{\frac{1}{2}i\mathrm{θ}}$

By equating each like terms, we get

$R=r^{\frac{1}{2}}$And$\mathrm{ϕ}=\frac{\mathrm{θ}}{2}$ or $2\mathrm{ϕ}=\mathrm{θ}$

So 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 $\mathrm{θ}=\mathrm{π}$ then the image of it will be the$\mathrm{ϕ}=\frac{\mathrm{π}}{2}$.

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

Hence, $R=r^{\frac{1}{2}}$and$2\mathrm{ϕ}=\mathrm{θ}$ .