Exponential and trigonometrical fittings: user-friendly expressions for the coefficients

The coefficients of numerical methods derived by exponential and trigonometric fittings are functions of parameter z = omega h where omega and h are the involved frequency and the step size, respectively. The problem is that, for the versions described until now in the literature, the accurate computation of each coefficient asks for four different formulas (an analytic formula valid for big z and a power series expansion for small z, in each of the two fittings). In this paper, we describe an algorithm-like technique which allows replacing the set of the four by a single formula. The latter is universally valid, in the sense that it can be successfully applied irrespective of whether z is small or big, or of whether the fitting is trigonometric or exponential. Two sets of special functions, sets C and S, are introduced for this purpose, and their mathematical properties are established. Examples and applications are also presented.