Abstract:
The problem of analytic continuation is considered for the Lauricella function F(N)DF(N)D, a generalized hypergeometric functions of NN complex variables. For an arbitrary NN a complete set of formulae is given for its analytic continuation outside the boundary of the unit polydisk, where it is defined originally by an NN-variate hypergeometric series. Such formulae represent F(N)DF(N)D in suitable subdomains of CN in terms of other generalized hypergeometric series, which solve the same system of partial differential equations as F(N)D. These hypergeometric series are the N-dimensional analogue of Kummer's solutions in the theory of Gauss's classical hypergeometric equation. The use of this function in the theory of the Riemann–Hilbert problem and its applications to the Schwarz–Christoffel parameter problem and problems in plasma physics are also discussed.
Bibliography: 163 titles.
Keywords:
multivariate hypergeometric functions, systems of partial differential equations, analytic continuation, Riemann–Hilbert problem, Schwarz–Christoffel integral, crowding problem, magnetic reconnection phenomenon.
This research was carried out with the support of the Peoples' Friendship University of Russia, programme “5-100”,
and the Russian Foundation for Basic Research (grant nos. 16-01-00781 and 16-07-01195).
Citation:
S. I. Bezrodnykh, “The Lauricella hypergeometric function F(N)D, the Riemann–Hilbert problem, and some applications”, Russian Math. Surveys, 73:6 (2018), 941–1031
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\by S.~I.~Bezrodnykh
\paper The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann--Hilbert problem, and some applications
\jour Russian Math. Surveys
\yr 2018
\vol 73
\issue 6
\pages 941--1031
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