I'm reading Griffiths's Introduction to Quantum Mechanics 3rd ed textbook [1]. On p.136, the author explains:
But wait! Equation 4.25 (angular equation for the $\theta$-part) is a second-order differential equation: It should have two linearly independent solutions, for any old values of $\ell$ and $m$. Where are all the other solutions? (One is related to the associated Legendre function.) Answer: They exist of course, as mathematical solutions to the equation, but they are physically unacceptable because they blow up at $\theta=0$ and/or $\theta=\pi$ (see Problem 4.5).
In problem 4.5, I can find that the function $A\ln[\tan (\theta/2)]$ satisfies the $\theta$ equation for $\ell=m=0$. And such function blows up at $\theta=0$ and $\theta=\pi$.
But why such function is physically unacceptable? For the wave function to be physically acceptable, it fundamentally needs to be square-integrable. And $\ln[\tan (\theta/2)]$ does actually!
$$\int_{0}^\pi [\ln[\tan (\theta/2)]]^2\sin\theta \text d\theta= \frac{\pi^2}6$$
For the well-behaved function case, it makes sense to set the function condition 'finite' and 'square-integrable' equivalently. In this case, although $\ln[\tan (\theta/2)]$ blows up at $\theta=0$ and $\theta=\pi$, it is still square-integrable tamed by $\sin \theta$ term. So it can be normalized to satisfy the Born's statistical interpretation. But the author says such function is physically unacceptable so I wonder why.
Reference
Griffiths, D. J.; Schroeter, D. F. Introduction to Quantum Mechanics 3rd ed; Cambridge University Press, 2018. ISBN 978-1107189638.