Oh sorry about my unfinished answer:
A function from some subset of R to R (real numbers) is analytic on an open set S if the function has a power
series (convergent in the usual meaning) on S.
The wikipedia article (http://en.wikipedia.org/wiki/Analytic_function) is OK, I think.
Is an expression that involves the Lauricella function (http://en.wikipedia.org/wiki/Lauricella_hypergeometric_series)
closed form? Oh, few would say it is, but this is largely just tradition.
--Barton
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A function from some subset of C to C (complex numbers) is analytic on an open set S if the function is differentiable on S.
A function from some subset of R to R (real numbers) is analytic on an open set S if the