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## Normalization for final result

Define , then by separation of variables:
 (3.14)

Introduce . From our previous discussion, we find that,

 (3.15)

Or just as easily,

Here we need to compensate for the following: The z used in this previous equation was built with:

 (3.16)

Thus we must shift . Thus,

We can find as another separation of variables chain:
 (3.17)

The non-integral part equals zero, the integral component cycles up on the n derivative and down on the m derivative. Of course, , but we maintain the separation for the sake of derivation. The final result of the cycle gives:

 (3.18)

To solve the latter integral, we introduce two identities:
 (3.19)

We follow through this derivation cycle n times, splitting each P value as per the above identity. The integral becomes the as identified above and he array will be spread from values of . If we integrate both sides of the equation over the Legendre function range, [-1,1], all said values of will vanish except . The prefactors from the expansion give:
 (3.20)

But properly,
 (3.21)

From this we conclude,

 (3.22)

Of course, we cannot neglect the symmetric components of theta, which give us an additional normalization of , thus the final normalization condition,

 (3.23)

Re-tagging our variables appropriately, the normalized Associated Legendre Polynomial is,

Or in simplified notation,

 (3.24)

.

Next: Radial side Up: The Angular Component Previous: A closer look
tim jones 2009-02-11