Numerical evaluation of analytic functions by Cauchy's theorem
- 271 Downloads
The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.
AMS Categoriesprimary: 65E05 secondary: 30E20, 65D32
Key-words and phrasesanalytic functions asymptotic estimates Cauchy formula Cauchy theorem circle contour complex contour integrals error bounds error term numerical integration Taylor series trapezoidal quadrature rule
Unable to display preview. Download preview PDF.
- 1.R. V. Churchill, J. W. Brown and R. F. Verhey,Complex Variables and Applications, 3rd ed., McGraw-Hill, New York (1974), 115–121, 129–133, 145–147.Google Scholar
- 2.P. J. Davis and P. Rabinowitz.Methods of Numerical Integration, 2nd ed., Academic Press, Orlando, Florida (1984), 168–172.Google Scholar
- 3.L. M. Delves and J. N. Lyness,A numerical method for locating the zeros of an analytic function, Math. Comp. 21 (1967), 543–560.Google Scholar
- 4.D. Elliott and J. D. Donaldson,On quadrature rules for ordinary and Cauchy principal value integrals over contours, SIAM J. Numer. Anal. 14 (1977), 1078–1087.Google Scholar
- 5.R. J. Fornaro,Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal. 10 (1973), 623–634.Google Scholar
- 6.N. I. Ioakimidis,A modification of the classical quadrature method for locating zeros of analytic functions, BIT 25 (1985), 681–686.Google Scholar
- 7.J. N. Lyness and L. M. Delves,On numerical contour integration round a closed contour, Math. Comp. 21 (1967), 561–577.Google Scholar
- 8.J. N. Lyness and G. Sande,Algorithm 413: ENTCAF and ENTCRE: Evaluation of normalized Taylor coefficients of an analytic function, Comm. ACM 14 (1971), 669–675. [Also in:Collected Algorithms from ACM, Vol. II:Algorithms 221–492, The Association for Computing Machinery (ACM), New York (1980), 413-P1-0–413-P7-0.]Google Scholar
- 9.E. N. Theotokoglou,Modified trapezoidal quadratures and accurate potential evaluation for numerical solutions of S.I.E.'s on the unit circle, Internat. J. Numer. Methods Engrg. 26 (1988), 2115–2128.Google Scholar