User:Dnessett/Legendre/Associated Legendre Functions Orthogonality for fixed m

[This page is deprecated in favor of User:Dnessett/Legendre/Associated Legendre Functions Orthonormality for fixed m, since the theorem not only shows the orthogonality of the Associated Legendre Functions, but also provides the normalization constant. All future modifications will be made to the referenced page.]

This article provides a proof that the associated Legendre functions are orthogonal for fixed m.

Theorem

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[Note: This article uses the more common   notation, rather than  ]

Where:    

Proof

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The Associated Legendre Functions are regular solutions to the general Legendre equation:   , where  

This equation is an example of a more general class of equations known as the Sturm-Liouville equations. Using Sturm-Liouville theory, one can show that   vanishes when   However, one can find   directly from the above definition, whether or not  

 

Since k and occur symmetrically, one can without loss of generality assume that   Integrate by parts   times, where the curly brackets in the integral indicate the factors, the first being   and the second   For each of the first   integrations by parts,   in the   term contains the factor  ; so the term vanishes. For each of the remaining integrations,   in that term contains the factor  ; 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   (remembering that   ). The other derivative is non-zero only when  , that is, when   Because   these two conditions imply that the only non-zero term in the sum occurs when   and   So:

 

To evaluate the differentiated factors, expand   using the binomial theorem:   The only thing that survives differentiation   times is the   term, which (after differentiation) equals:  . Therefore:

  ................................................. (1)

Evaluate   by a change of variable:

 

Thus,   [To eliminate the negative sign on the second integral, the limits are switched from   to   , recalling that   and  ].

A table of standard trigonometric integrals shows:   Since     for   Applying this result to   and changing the variable back to   yields:   for   Using this recursively:

 

Applying this result to (1):

  QED.

See also

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References

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  • Kenneth Franklin Riley, Michael Paul Hobson, Stephen John Bence, "Mathematical methods for physics and engineering", (2006) 3  Edition, Cambridge University Press, ISBN 0-521-67971-0.