Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

Abstract

Abstract. Herein, it is shown that by exploiting integral definitions of well known special functions, through generalizations and differentiations, broad classes of definite integrals can be solved in closed form or in terms of special functions. This is especially useful when there is no closed form solution to the indefinite form of the integral. In this paper, three such classes of definite integrals are presented. Two of these classes incorporate and supercede all of K61big's integration formulae [11], including his formulation for the computation of Cauchy principal values. Also presented are the mathematical derivations that support the implementation of a third class which exploits the incomplete Gamma function. The resulting programs, based on pattern matching, differentiation, and occasionally limits, are very efficient.

Instantaneous Mutual Information and Eigen-Channels in MIMO Mobile Rayleigh Fading

“…by inserting (36) and (35) into (38), and (36), (35) and (37) into (39), respectively. In general, it seems difficult to further simplify (36), (35) and (37).…”

“…However, if ν is small compared to M , the gap between the low-and high-SNR approximations is large (see the results for the 4 × 4 system). Therefore, we need to resort to the exact formulas in (36), (35), (37) and (39) to calculate the accurate values of the correlation coefficient, for not so small or large SNRs. For example, at η = 15 dB, the simulation and exact theoretical curves, as well as lowand high-SNR approximations are shown in Fig.…”

Exchange Energy for Two-Active-Electron Diatomic Systems Within the Surface Integral Method

“…Although the analysis was done for the simple case c 1 = c 2 , for c 1 = c 2 , one can always return to the roots of (19). Thus the twice transformed integral of (18) has the simple and recognizable form:…”

“…After decoupling the correlation term ρ 12 in terms of ρ i where i = 1, 2 and transforming to cylindrical coordinates (as discussed in section 3), we find that the resulting integrals in ρ i are exactly solvable [19] and implemented in various computer algebra systems.…”

On a bessel function integral

“…Secondly, the extension of the analytical result allows us to express solutions of a class of definite integrals involving products of Bessel functions, in terms of Bessel functions and closed form expressions. Since these integrals are of a type often encountered in the study of waves in thermal plasmas and other media, their inclusion is warranted into the scheme of definite integration outlined and discussed in previous work [4,3].…”