Solving the wave equation for one-electron atom

Not sure what more could be added. Think about what would happen if that function was NOT single-valued. A multi-valued function is not differentiable.

Once you are fine with that condition apply it to the wave function.

$$exp(i m phi) = exp(i m (phi + 2 pi));$$

This is the mathematical expression of single valuedness for the function, also called periodicity since we arrive at the same location for different values of phi that differ by exactly $2 pi$.

This reduces to

$$1 = exp(i 2 m pi)$$

The only possible solution are $m = rm integer$.

If you are not familiar with Euler's identity the above equation is the same as

$$1 = cos(2 m pi) + i sin(2 m pi)$$

This requires

$$cos(2 m pi) = 1$$

$$sin(2 m pi) = 0 $$

Here $phi$ is the polar angle, i.e. it changes from $0$ to $2pi$. Any change greater than that takes us a full rotation around the center of the coordinates, to the point where we have already been, i.e. we should get the same value of the wave function. Mathematically this may be expressed as: begin{equation} Phi(phi + 2pi) = Phi(phi). end{equation}

If we now take the solution in the form $Phi(phi) = e^{ilambdaphi}$, then begin{equation} Phi(phi + 2pi) = e^{ilambda(phi+2pi)} = e^{ilambdaphi} = Phi(phi), end{equation} which implies that begin{equation} e^{ilambda 2pi} = 1. end{equation} This is possible, only if $lambda$ is an integer number.