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Heun differential equations are the most general second order Fuchsianequations with four regular singularities. An explicit integral seriesrepresentation of Heun functions involving only elementary integrands hashitherto been unknown and noted as an important open problem in a recentreview. We provide explicit integral representations of the solutions of allequations of the Heun class: general, confluent, bi-confluent, doubly-confluentand triconfluent, with integrals involving only rational functions andexponential integrands. All the series are illustrated with concrete examplesof use. These results stem from the technique of path-sums, which we use toevaluate the path-ordered exponential of a variable matrix chosen specificallyto yield Heun functions. We demonstrate the utility of the integral series byproviding the first representation of the solution to the Teukolsky radialequation governing the metric perturbations of rotating black holes that isconvergent everywhere from the black hole horizon up to spatial infinity.

Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory