Scattering of laser radiation under conditions of optical breakdown of transparent dielectrics and gases

Abstract

By employing certain concepts in combinatorial analysis, we derive here the closed-form expressions for the nth-order derivatives of the following: (i) inverse powers of a matrix d (ii) reciprocal powers of a function f (explicit or implicit) (iii) reciprocal powers of a determinant A and establish their relation with the Bell polynomials.