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• main points understood, and expand them - what is your understanding of what the points were.
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Mathematical method to get Laplacian

• needs to be finished

Legendre Polynomials

<math>\Theta(\theta):\alpha,\beta = m^2</math>

where <math>\alpha</math> is separation of the r-dependent part and <math>\beta</math> is the separation of the <math>\phi</math> dependent part.

Then the Differential equation form of <math>\Theta(\theta)= P_l^m (cos\theta)</math> replacing <math>z=cos\theta</math>

is: <math>(1-z^2)\frac{d^2U}{dz^2} - 2z \frac{dU}{dz} + \alpha U = 0</math>

Then we can take

<math>U(z)= \sum_{n=0}^\infty a_n z^n </math>

We can then take <math>\xi=\frac{1-z}{2}</math> where <math>-1 < z < 1 </math> and therefore <math>0 < z < 1</math> Then <math>U(\xi)</math> is also a differential equation where

<math>U(\xi)=\sum_{n=0}^\infty a_n \xi^2</math>

From this we get

<math> const. U'' + const. U' + const. U= const</math> for all <math>\xi</math>

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