The Bell System Technical Journal 



Vol. XV July, 1936 No. 3 



A Laplacian Expansion for Hermitian-Laplace Functions of 



High Order* 



By E. C. MOLINA 



Among the wide variety of practical and theoretical problems con- 

 fronting the telephone engineer, there is a surprisingly large number to 

 whose solution mathematics has made notable contribution. In his kit of 

 mathematical tools the theory of probability is a frequently used and most 

 effective instrument. This theory of probability contains a large number 

 of theorems, a large number of functions, which permit of application to 

 telephony. Among these is a particular tool, a particular group of mathe- 

 matical functions known as the "Hermitian Functions," each of which is 

 identified by a number called its "order." These mathematical functions 

 or relations have no practical utility until the variables in the equation can 

 be assigned numerical values and the resultant numerical value of the 

 function calculated. Tables of the numerical values of Hermitian functions 

 of low order exist; for example. Glover's Tables of Applied Mathematics 

 cover the ground for those of the first eight orders. But tables for the 

 functions of higher order are still a desideratum. This paper presents an 

 expansion by means of which the evaluation of a high order function can 

 be readily accomplished with a considerable degree of accuracy. 



The development of the expansion is prefaced by some remarks on the 

 early history of the Hermitian functions and the relation of this history 

 to modern theoretical physics. 



I 



AMONG contributions made by Laplace to the domain of pure and 

 applied mathematics, two of great practical value are : 



(a) His method of evaluating definite integrals ^ whose integrands 



involve factors raised to high powers; 



(b) The pair of orthogonal polynomial functions ^ which he defined by 



the following Equations (1) and (2) 



(1) l{2n)l^[^/2^^n\']Un(u) = H e-'^^x - iuY^dx 



*J — 00 



/^oo 



= 2e"^ I e-^V" cos {2ux)dx\ 



(2) [(2« + l)!VW22"w!]^7„'(w) = i e-^\x - iuY^+Hx 



/»00 



= 2e"" 1 g-^Vn+i sin (2ux)dx. 



* Presented at International Congress of Mathematicians, Oslo, Norway, July 

 13-18, 1936. 



355 



