On the Representation of Transients by Series of Orthogonal Functions

When problems having to do with transients are solved by the Laplace transform or equivalent methods, one may be left with the necessity of solving a rather complicated equation in the transform variable. This may be avoided, in many cases, by getting the solution in the form of a series. Laguerre functions have had some use for that purpose. It is shown here how another set of functions, which are just the Jacobi polynomials whose argument is an exponential, may be used instead. The use of this latter set permits a rather elegant means of evaluating the coefficients in the expansion to be used. In an appendix, ways of applying the mathematical techniques used are investigated. These involve the complex "Faltung" theorem, for investigating questions of orthogonality and orthonormality in general.