Associated Legendre function/Proofs

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More information relevant to Associated Legendre function.

It will be demonstrated that the associated Legendre functions are orthogonal and their normalization constant will be derived.

Theorem

where:

Proof

The associated Legendre functions are regular solutions to the associated Legendre differential equation given in the main article. The equation is an example of a more general class of equations known as the Sturm-Liouville equations. Using Sturm-Liouville theory, one can show the orthogonality of functions with same superscript m and different subscripts:

In the case k = l it remains to find the normalization factor of the associated Legendre functions such that the "overlap" integral

One can evaluate the overlap integral directly from the definition of the associated Legendre polynomials given in the main article, whether or not k = l. Indeed, insert twice the definition:

Since k and l occur symmetrically, one can without loss of generality assume that l ≥ k. Use the well-known integration-by-parts equation

l + m times, where the curly brackets in the integral indicate the factors, the first being u and the second v’. For each of the first m integrations by parts, u in the term contains the factor (1−x2), so the term vanishes. For each of the remaining l integrations, v in that term contains the factor (x2−1) so the term also vanishes. This means:

Expand the second factor using Leibnitz' rule:

The leftmost derivative in the sum is non-zero only when r ≤ 2m (remembering that ml). The other derivative is non-zero only when k + l + 2mr ≤ 2k, that is, when r ≥ 2m + lk. Because lk these two conditions imply that the only non-zero term in the sum occurs when r = 2m and l = k. So:

where δkl is the Kronecker delta that shows the orthogonality of functions with lk. To evaluate the differentiated factors, expand (1−x²)k using the binomial theorem:

The only term that survives differentiation 2k times is the x2k term, which after differentiation gives

Therefore:

Evaluate

by a change of variable:

Thus,

where we recall that

The limits were switched from

,

which accounts for one minus sign and further for integer l: (−1)2l =1 . Integration of

gives

Since

it follows that

for n > 1.

Applying this result to

and changing the variable back to x yields:

for l ≥ 1. Using this recursively:

Applying this result to equation (1):

Clearly, if we define new associated Legendre functions by a constant times the old ones,

then the overlap integral becomes,

that is, the new functions are normalized to unity.

Comments

The orthogonality of the associated Legendre functions can be demonstrated in different ways. The proof presented above assumes only that the reader is familiar with basic calculus and it is therefore accessible to the widest possible audience. However, as mentioned, their orthogonality also follows from the fact that the associated Legendre equation belongs to a family known as the Sturm-Liouville equations.

It is possible to demonstrate their orthogonality using principles associated with operator calculus. Let us write

where

Clearly, the case m = 0 is,

The proof given above starts out by implicitly proving the anti-Hermiticity of ∇. Indeed, noting that w(x) is a function with w(1) = w(−1) = 0 for m ≠ 0, it follows from partial integration that

Hence

To demonstrate orthogonality of the associated Legendre polynomials, we use a result from the theory of orthogonal polynomials. Namely, a Legendre polynomial of order l is orthogonal to any polynomial Πp of order p lower than l. In bra-ket notation

Knowing this,

The bra is a polynomial of order k, because

where it was used that m times differentiation of a polynomial lowers its order by m and that the order of a product of polynomials is the product of the orders. Since we assumed that kl, the integral is non-zero only if k = l. Hence it follows readily that the associated Legendre polynomials of equal superscripts and non-equal subscripts are orthogonal. However, the hard work (given above) of computing the normalization for the case k = l remains to be done.