In a recent paper [Phys. Lett. A 205, 130 (1995)], we investigated the inverse problem of solving g([Formula Presented],...,[Formula Presented]) from the integral equation n([Formula Presented],...,[Formula Presented])=∫K([Formula Presented],...,[Formula Presented][Formula Presented],...,[Formula Presented])g(x [Formula Presented],...,[Formula Presented])[Formula Presented]...[Formula Presented], with the given integral n and kernel K by analytically dilating variable y to the complex plane. We showed, by studying the singularities and discontinuities of the dilated kernel and integral, that the unknown function g can be obtained from an algebraic relation in the case where the dilated kernel contains a simple and single-valued pole. The present paper intends to generalize this result to the case where the kernel contains higher-order and/or multivalued poles. We show that the integral equation in these more general cases can be transformed to algebraic, ordinary, or partial differential equations, depending on the type of the singularities of the kernel and the dimension of the inverse problem. Moreover, some conditions constraining the integral n, which are independent of the integrand g, are revealed when K has multivalued or high-order singularities.

Original languageEnglish (US)
Pages (from-to)2384-2391
Number of pages8
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Issue number3
StatePublished - 1996

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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