The angular equation:(adsbygoogle = window.adsbygoogle || []).push({});

##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta##

Right now, ##l## can be any number.

The solutions are Legendre polynomials in the variable ##\cos\theta##:

##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer.

Why must ##l## be a non-negative integer?

In the case of the solution to Laplace's equation in cartesian coordinates, ##k=\frac{n\pi}{a}## because we need to satisfy the boundary conditions. How about in the case of spherical coordinates?

Does it mean that if ##l## is not a non-negative integer, but is, say, ##\sqrt3##, then the differential equation can't be solved?

That is, the equation ##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=(-3-\sqrt3)\sin\theta\,\Theta## has no solution?

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# Solving Laplace's equation in spherical coordinates

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