(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) |

So we follow through,

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) |

.