Solving the laplace equation in Spherical coordinates

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Solutions to this equation are called harmonics

Using the definition for the laplace operator in spherical coordinates, it follows:

Using sepration of variables becomes

Applying this to the original equation produces

Because the terms depend on different independant variables, the only way the equation holds is if both terms are constant.

We again use separation of variables to solve the partial differential equation of the angle dependant term.

Replacing into the equation

Based on a similar argument it follows that both terms must be constant, with this we may now solve for

Solve using Frobenius Method