An Expansion for Laplacian Integrals in Terms of Incomplete Gamma Functions, and Some Applications

L APLACE has given us, in the Theorie Analytique des Probabilities, Book I, Part II, Chapter I, a method of approximating by means of series to the value of a definite integral of the type I = j f , v > · · · yn6»dx, where y, yz, · · · yn are functions of x whose exponents dh 02, · · · 0n are of the same order of magnitude as a large number 6. The last function in the integrand, embraces all factors whose exponents are of low order of magnitude compared with 6. The integral here considered must not be confused with the well known "Laplacian Transform" integral which is also embodied in the Theorie Analytique. When the function >(x) is other than a mere constant, the Laplacian method as presented in the Theorie Analytique does not, in certain cases, give us a series which reduces to its first term as 6 approaches infinity. The object of this paper is to present a modification of the method which gives the desired result and to present two applications of considerable practical importance. The modification consists in divorcing the function 0 from the factors of the integrand raised to high powers and associating with the factor (dx/dt) which makes its appearance with the Laplacian change of variable from re to t. DEDUCTION OF MODIFIED EXPANSION