It is demonstrated that the associated Legendre functions are orthogonal in two different ways and their normalization constant for each is derived.
Theorem (orthogonality relation 1)
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 the same superscript m and different subscripts:
However, one can find
directly from the above definition, whether or not
This involves evaluating the overlap integral directly from the definition of the associated Legendre functions given in the main article. Indeed, inserting the definition of the function twice:
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 m ≤ l). The other derivative is non-zero only when k + l + 2m − r ≤ 2k, that is, when r ≥ 2m + l − k. Because l ≥ k these two conditions imply that the only non-zero term in the sum occurs when r = 2m and l = k.
where δkl is the Kronecker delta that shows the orthogonality of functions with l ≠ k. The factor at the front of comes from switching the sign of x2-1 inside (x2-1)l.
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
by a change of variable:
where we recall that
The limits were switched from
which accounts for one minus sign .
it follows that
for n > 1.
Applying this result to
and changing the variable back to x
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.
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
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
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
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 k ≤ l, 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.
Theorem (orthogonality relation 2)
The functions (x2-1)k are even functions; so their
jth order derivatives are even or odd functions according as j is even or odd.
Therefore, if m or n is even but the other is odd, then one of the two factors in curly braces in the
preceding expression is an even function, and the other is an odd function. This makes the entire
integrand an odd function. When integrated between the limits that are negatives of each other,
it yields 0, as it should, since . So we need to consider further
only even integrands, for which m and n are either both even or both odd. In this case all exponents
in the integrand are non-negative integers (except when m = n = 0).
The integrand can be integrated by parts times, the first
factor in curly braces and its derivatives being identified as
u and the second factor as v' in the formula:
Since m and n occur symmetrically, we can assume without loss
of generality that . For the first n-1 integrations
by parts, the uv term vanishes at the limits because u includes
at least one factor of (1-x2). The same also is true for the
nth integration, unless n = m, in which case the uv term is:
For the remaining integrations the uv term vanishes at the
limits because v includes at least one factor 1-x2 (and perhaps u does also). The result is:
The highest power of x in the binomial expansion of (x2-1)l
is ; after derivatives of it are taken, the highest power
is . The highest power of x in the expansion of
is m+n-2; so the highest power in the expression in curly braces
is . After derivatives of it are taken, the highest
power is 0; that is, the expression is a constant. Taking the
remaining 2 derivatives causes the integrand to vanish. Therefore:
Since this function is odd, we need evaluate it only at its upper
limit x = 1 and double the result.
An expression of the form can be differentiated using Leibnitz's rule. Only one term in the sum is of interest here, namely the one in which (x-1)k is differentiated exactly k times so that no factors of (x-1) remain. If it is differentiated fewer times, then it vanishes at x = 1. If it is differentiated more times, then it vanishes everywhere. So, ignoring terms with factors of (x-1), we have (from right to left in the preceding expression):
, which equals when x = 1:
- , which equals when x = 1.
which has been doubled to include the lower limit x = -1. QED.
When n = m = 0:
When expanded in powers of (x+1) and (x-1) by using Leibnitz's
rule, the expression in curly braces includes exactly one term,
(x-1)l, having no factor (x+1) and exactly one term, (x+1), having no factor (x-1). Therefore in a Laurent series expansion of the integrand about either x = 1 or x = -1 all terms except
one have non-negative exponents, and their integral is finite.
The one remaining term in the integrand is a constant multiple
of 1/(x+1) or 1/(x-1); so its integral logarithmically diverges
at the limits, making T infinite.