# Orthogonality of Associated Legendre Functions for Noninteger Order and Index

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This Demonstration shows an unusual orthogonality relation for Legendre functions of the first kind . In contrast to the well-known situation of integer degree and integer order (associated Legendre polynomials), in this case both degree and order are allowed to take noninteger real values. For fixed , these functions satisfy an orthogonality relation, according to which is 0 whenever is equal to 2 if and , and is given by the formula in any other case.

Contributed by: Axel Schulze-Halberg and John R. Morris (May 2013)

Open content licensed under CC BY-NC-SA