Appell Series - Integral Representations

Integral Representations

The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn & Ryzhik 1971, § 9.184). However, Émile Picard (1881) discovered that Appell's F₁ can also be written as a one-dimensional Euler-type integral:

$F_1(a,b_1,b_2,c; x,y) = \frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \,\mathrm{d}t, \quad \real \,c > \real \,a > 0 ~.$

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

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