An Expansion for Laplacian Integrals in Terms of 



Incomplete Gamma Functions, and 



Some Applications* 



By EDWARD C. MOLINA 



Introduction 



LAPLACE has given us, in the Theorie Analytique des Probabilites, 

 Book I, Part II, Chapter I, a method of approximating by means 

 of series to the value of a definite integral of the type 





where Ti, V2, • • • yn are functions of x whose exponents 9i, 62, • • • On 

 are of the same order of magnitude as a large number 6. The last 

 function in the integrand, 4>, embraces all factors whose exponents 

 are of low order of magnitude compared with d. 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 0(.r) 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 (/> from the factors of the integrand raised to 

 high powers and associating <^ with the factor {dxjdt) which makes its 

 appearance with the Laplacian change of variable from x to /. 



Deduction of Modified Expansion 

 Setting dsid = Ys we have 



3'/'3'2'' • • • 3'.'" = (>'i^'3'2'' • • • yn'-y= /, 



(1) 1= r[y{x)Jcf>(x)dx, 



which we shall write in the form 



1= r \:y(xr4>u>ix)dxj 



* Presented by title at International Congress of Mathematicians, Zurich, Swit- 

 zerland, September, 1932. 



563 



