Applications of Hilbert transform theory to numerical quadrature

Authors:
W. E. Smith and J. N. Lyness

Journal:
Math. Comp. **23** (1969), 231-252

MSC:
Primary 65.55; Secondary 44.00

DOI:
https://doi.org/10.1090/S0025-5718-1969-0251906-9

MathSciNet review:
0251906

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Some finite integrals are difficult to evaluate numerically because the integrand has a high peak or contains a rapidly oscillating function as a factor. If the integrand is an analytic function Cauchy's theorem may be applied to replace the integral by a contour integral, the path being chosen to avoid singularities of the integrand, together with a possible residue contribution. If the integrand has branch singularities in the complex plane close to the interval of integration, the direct application of Cauchy's theorem is not practical. In this paper we show how the theory of Hilbert transforms may be applied to replace the integrand by a different complex valued function whose real part coincides with the integrand on the real line, but which has no singularities in the upper half plane. Using these transformations, integrands whose difficult behavior arises from a factor whose Hilbert transform is known analytically may be treated by carrying out a contour integral of a different function and taking the real part of the result. It is shown by means of examples that such a procedure may result in significant savings in terms of computational effort.

**[1]**M. Abramowitz, ``On the practical evaluation of integrals,''*J. Soc. Indust. Appl. Math.*, v. 2, 1954, pp. 20-35.MR**15**, 992. MR**0062517 (15:992a)****[2]**M. Abramowitz & I. A. Stegun, (Editors),*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, Nat. Bur. Standards Appl. Math. Ser., 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964; 3rd printing, with corrections, 1965. MR**29**#4914; MR**31**#1400. MR**0167642 (29:4914)****[3]**F. L. Bauer, H. Rutishauser & E. Stiefel,*New Aspects in Numerical Quadrature*, Proc. Sympos. Appl. Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1963, pp. 199-218. MR**30**#4384. MR**0174177 (30:4384)****[4]**P. J. Davis & P. Rabinowitz,*Numerical Integration*, Blaisdell, Waltham, Mass., 1967. MR**35**#2482. MR**0211604 (35:2482)****[5]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Tables of Integral Transforms*. Vols. I, II, McGraw-Hill, New York, 1954, 1955. MR**15**, 868; MR**16**, 468. MR**0061695 (15:868a)****[6]**T. Havie, ``On a modification of Romberg's algorithm,''*Nordisk Tidskr. InformationsBehandling*, v. 6, 1966, pp. 24-30. MR**33**#3460. MR**0195257 (33:3460)****[7]**J. R. Macdonald & M. K. Brachman, ``Linear-system integral transform relations,''*Rev. Mod. Phys.*, v. 28, 1956, pp. 393-422. MR**18**, 652. MR**0083063 (18:652c)****[8]**W. M. McKeeman, ``Algorithm 145; adaptive numerical integration by Simpson's rule,''*Comm. ACM*, v. 5, 1962, p. 604. See also: ``Certification of algorithm 145; adaptive numerical integration by Simpson's rule,''*Comm. ACM*, v. 6, 1963, pp. 167-168.**[9]**N. I. Muskhelishvili,*Singular Integral Equations, Boundary Problems of Function Theory and Their Application to Mathematical Physics*, OGIZ, Moscow, 1946; English transl., Noordhoff, Groningen, 1953. MR**8**, 586; MR**15**, 434. MR**0058845 (15:434e)****[10]**F. G. Tricomi,*Integral Equations*, Interscience, New York, 1957. MR**20**#1177. MR**0094665 (20:1177)****[11]**J. V. Uspensky, ``On the convergence of quadrature formulas related to an infinite interval,''*Trans. Amer. Math. Soc.*, v. 30, 1928, pp. 542-559. MR**1501444**

Retrieve articles in *Mathematics of Computation*
with MSC:
65.55,
44.00

Retrieve articles in all journals with MSC: 65.55, 44.00

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1969-0251906-9

Article copyright:
© Copyright 1969
American Mathematical Society