Uncoupling the differential equations arising from a technique for evaluating indefinite integrals containing special functions or their products
Author:
Jean C. Piquette
Journal:
Quart. Appl. Math. 48 (1990), 95-112
MSC:
Primary 33C55; Secondary 68Q40
DOI:
https://doi.org/10.1090/qam/1040236
MathSciNet review:
MR1040236
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Abstract: A previous article by Piquette and Van Buren [1] described an analytical technique for evaluating indefinite integrals involving special functions or their products. The technique replaces the integral by an inhomogeneous set of coupled first-order differential equations. This coupled set does not explicitly contain the special functions of the integrand, and any particular solution of the set is sufficient to obtain an analytical expression for the indefinite integral. It is shown here that the coupled set arising from the method always occurs in normal form. Hence, it is amenable to the method of Forsyth [6] for uncoupling such a set. That is, the solution of the set can be made to depend upon the solution of a single differential equation of order equal to the number of equations in the set. Any particular solution of this single equation is then sufficient to yield the desired indefinite integral. As examples, the uncoupled equation is given here for integrals involving (i) the product of two Bessel functions, (ii) the product of two Hermite functions, or (iii) the product of two Laguerre functions, and a tabulation of integrals of these types is provided. Examples involving products of three or four special functions are also provided. The method can be used to extend the integration capabilities of symbolic-mathematics computer programs so that they can handle broad classes of indefinite integrals containing special functions or their products.
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D. Withoff, Algorithm Development Dept., Wolfram Research, Inc. (private communication)
J. C. Piquette and A. L. Van Buren, Technique for evaluating indefinite integrals involving products of certain special functions, SIAM J. Math. Anal. 15 (4), 845–855 (1984)
N. J. Sonine, Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries, Math. Ann. 16, 1–80 (1880)
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London, 1966, pp. 132–134
C. Truesdell, An Essay Toward a Unified Theory of Special Functions Based Upon the Functional Equation $\frac {{\partial F\left ( {z, \alpha } \right )}}{{\partial z}} = F\left ( {z, \alpha + 1} \right )$, Princeton University Press, Princeton, 1948
J. C. Piquette, An analytical expression for coefficients arising when implementing a technique for indefinite integration of products of special functions, SIAM J. Math. Anal. 17 (4), 1033–1035 (1986)
A. R. Forsyth, Theory of Differential Equations, Dover, New York, 1959, Vol. II, pp. 18–19
S. L. Ross, Introduction to Ordinary Differential Equations, Wiley, New York, 1980, p. 268, Eq. (7.5)
E. C. Lommel, Zur Theorie der Bessel’schen Functionen, Math. Ann. 14, 510–536 (1879)
L. C. Maximon, On the evaluation of indefinite integrals involving the special functions: Application of method, Quart. Appl. Math. 13, 84–93 (1955)
Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962, p. 257, Eq. (21), and p. 258, Eq. (23)
G. Arfken, Mathematical Methods for Physicists, Academic, New York, 1971, pp. 613–615
J. C. Piquette, Applications of a technique for evaluating indefinite integrals containing products of the special functions of physics, SIAM J. Math. Anal. 20 (5), 1260–1269 (1989)
E. Butkov, Mathematical Physics, Addison-Wesley, Reading, MA, 1968, pp. 338–339
N. N. Lebedev, Special Functions and Their Applications (R. A. Silverman, trans.), Prentice-Hall, Englewood Cliffs, NJ, 1965, p. 273
L. C. Maximon and G. W. Morgan, On the evaluation of indefinite integrals involving special functions: Development of method, Quart. Appl. Math. 13, 79–83 (1955)
G. M. Muller, On the indefinite integrals of functions satisfying homogeneous linear differential equations, Proc. Amer. Math. Soc. 5, 716–719 (1954)
Yu. F. Filippov, Tables of Indefinite Integrals of Higher Transcendental Functions, Vishcha Shkola, Kharkov, 1983
E. C. Lommel, Ueber eine mit den Bessel’schen Functionen verwandte Function, Math. Ann. 9, 425–444 (1876)
D. Withoff, Algorithm Development Dept., Wolfram Research, Inc. (private communication)
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© Copyright 1990
American Mathematical Society
