Neumanns Formula for the Legendre Functions

Neumann s Formula for the Legendre Functions. Let us now consider the integral 

From equation (15.9), it follows that the integrals in this series are zero if s , nr if s = n -)- 2r where r is a positive integer, so that the above series is equivalent to 

Certain related formulae, due to MacRobcrt, follow readily from this result. If j /t 1 and m is a positive integer then 

If m it follows from equation (15.9) that each term of this scries vanishes so that we have 

On the other hand if m = n -f- 1, the series (18.2) reduces to the single term 

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