Article

Applicable Algebra in Engineering, Communication and Computing

, Volume 1, Issue 2, pp 149-165

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

  • K. O. GeddesAffiliated withDepartment of Computer Science, University of Waterloo
  • , M. L. GlasserAffiliated withDepartment of Physics and Department of Mathematics and Computer Science, Clarkson University
  • , R. A. MooreAffiliated withGuelph-Waterloo Program for Graduate Work in Physics, Waterloo Campus
  • , T. C. ScottAffiliated withGuelph-Waterloo Program for Graduate Work in Physics, Waterloo Campusthe Institute for Theoretical Atomic and Molecular Physics at the Harvard—Smithsonian Center for Astrophysics

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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 Kölbig'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.

Keywords

Integration Symbolic Computation MAPLE Special Functions