Inverse problem and singularity of the integration kernel

Many important problems in physics and other sciences can be formulated in terms of the inverse problem of type n(y) = ∝K(y ∥ x)g(x) dx, where g(x) is unknown. We show that this problem can be completely solved for a quite general class of kernel K(y ∥ x) by analytically dilating n(y) and K(y ∥ x) to the complex z plane, and by the analysis of the singularity of the dilated kernel K(z ∥ x). The formalism is also extended to multi-dimensional cases.