A Method of Evaluating Laplace Transforms with Series of Complete or Incomplete Beta Functions,

In a previous paper factorial series were used to calculate ordinary and modified Bessel functions of the second kind. In the present paper the factorial series is generalized so that Laplace integrals in which the integrand has a branch point at the origin are represented by a series of beta functions. To effect the required transformation, formulas for calculating Stirling numbers of fractional order were derived these were used in the same manner as the Stirling numbers of integer order are used to calculate the coefficients of a factorial series. Formulas for calculating Kox and Klx have been derived and programmed, using these modified Stirling numbers. Formulas for calculating Iox and Ilx have been derived and programmed using series of incomplete beta functions in a similar algorithm. Results agree to thirteen significant figures for Kox and Klx when x 8 and for Iox when x 15. The modified Stirling numbers increase very slowly with order and index since gamma functions do not occur in the definition. Consequently no problems with overrun of the electronic computer occurred during the course of the calculations.