Integral representations for solutions of Heun's equation

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Abstract
In 1914 Whittaker(12) conjectured that the Heun differential equation is the simplest equation of Fuchsian type whose solution cannot be represented by a contour integral; instead the nearest approach to such a solution is to find a homogeneous integral equation satisfied by a solution of the differential equation. In this paper we reconsider Whittaker's conjecture and show that in fact solutions of Heun's equation can be represented in terms of contour integrals, similar to those of Barnes for the hypergeometric equation. The integrands of these integrals are of a rather complicated nature and cannot be said to involve known or simpler functions although they do provide expressions for the analytic continuation of Heun functions analogous to those for the hypergeometric functions.

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