A laplacian expansion for Hermitian-Laplace functions of high order

Among the wide variety of practical and theoretical problems confronting 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 mathematical 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.