Associated Legendre Functions

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for in the range . Here, is a non-negative integer (known as the degree), and is an integer (known as the order) lying in the range . The functions themselves take the form

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which implies that

Assuming that , the satisfy the orthogonality condition

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where is a Kronecker delta symbol.

The associated Legendre functions of order 0 (i.e.,
) are called *Legendre polynomials*, and
are denoted the
: that is,
. It follows that^{}

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It can also be shown that

provided and .

All of the associated Legendre functions of degree less than 3 are listed below:

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