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Volume 9, Issue 4

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Edited by Paul Abbott

Closed Form Integral

In the last column, a closed form for the integral

was obtained. Here was the result, with the denominator simplified slightly by noting that .

Elmar Zeitler (zeitler@fhi-berlin.mpg.de) suggested an alternative approach: expanding as a Gaussian hypergeometric function (functions.wolfram.com/Polynomials/LegendreP/26/01/01/0001),

and using functions.wolfram.com/05.03.07.0004.01 to write

one has to compute

which can be done using integration by parts. Alternatively, evaluating the integral for , the pattern is clear.

One observes that

leading to an alternative form for , denoted .

Since for , we check that for .

Alternatively, expanding both and as Gaussian hypergeometric functions, the result is immediate.

In other words, we have demonstrated the identity



     
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