INVERSE PROBLEM WITH A DILATED KERNEL CONTAINING DIFFERENT SINGULARITIES

In a recent paper [Phys. Lett. A 205, 130 (1995)], we investigated the inverse problem of solving g(x(1),...,x(q)) from the integral equatio n n(y(1),...,y(q))=integral (y(1),...,y(q)\x(1),...,x(q))g(x(1),...,x( q))dx(1) ... dx(q), with the given integral rr and kernel K by analyti cally dilating variable y to the complex plane. We showed, by studying the singularities and discontinuities of the dilated kernel and integ ral, that the unknown function g can be obtained from an algebraic rel ation in the case where the dilated kernel contains a simple and singl e-valued pole. The present paper intends to generalize this result to the case where the kernel contains higher-order and/or multivalued pol es, We show that the integral equation in these more general cases can be transformed Co algebraic, ordinary, or partial differential equati ons, depending on the type of the singularities of the kernel and the dimension of the inverse problem, Moreover, some conditions constraini ng the integral n, which are independent of the integrand g, are revea led when K has multivalued or high-order singularities.