A continued fraction expansion, with a truncation error estimate, for Dawson's integral
Author:
J. H. McCabe
Journal:
Math. Comp. 28 (1974), 811816
MSC:
Primary 65D20
MathSciNet review:
0371020
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Abstract: A continued fraction expansion for Dawson's integral is considered and an estimate of the truncation errors of the convergents of this continued fraction is provided. The continued fraction is shown to provide a sequence of rational approximations to the integral which have good convergence for both small and large values of the argument.
 [1]
W. J. Cody, K. A. Paciorek & H. C. Thacher, "Chebyshev approximations for Dawson's integral," Math. Comp, v. 24, 1970, pp. 171178. MR 41 #2883. MR 0258236 (41:2883)
 [2]
H. G. Dawson,"On the numerical value of ," Proc. London Math. Soc., v. 29, 1898, pp. 519522.
 [3]
W. Fair, "Padé approximations to the solution of the Ricatti equation," Math. Comp., v. 18, 1964, pp. 627634. MR 29 #6630. MR 0169380 (29:6630)
 [4]
D. G. Hummer, "Expansion of Dawson's function in a series of Chebyshev polynomials," Math. Comp., v. 18, 1964, pp. 317319. MR 29 #2967. MR 0165687 (29:2967)
 [5]
E. Laguerre, "Sur le réduction en fractions continues d'une fraction qui satisfait à une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels," J. Math. Pures Appl., v. 1, 1885.
 [6]
B. Lohmander, & S. Rittsten, "Table of the function ,'' Kungl. Fysiogr. Sällsk. i Lund Forh., v. 28, 1958, pp. 4552, MR 20 #1427. MR 0094919 (20:1427)
 [7]
Y. Luke, "The Padé table and the method," J. Mathematical Phys., v. 37, 1958, pp. 110127. MR 20 #5558. MR 0099114 (20:5558)
 [8]
J. H. McCabe, Continued Fraction Methods With Applications to First Order Ordinary Differential Equations, Ph D. Thesis, Brunel University, 1971.
 [9]
J. H. McCabe, "An extension of the Padé table to include two point Padé approximations" (To appear.)
 [10]
J. H. McCabe & J. A. Murphy, "Continued fraction expansions about two points" (To appear.)
 [11]
J. A. Murphy, Rational Approximations to the Error Function, Internal project, Brunel University, 1966.
 [12]
H. C. Thacher, Computation of the Complex Error Function by Continued Fractions, Blanch Anniversary volume, B. Mond (editor), Aerospace Research Laboratories, Washington, D. C.
 [13]
P. Wynn, "The numerical efficiency of certain continued fraction expansions. IB," Nederl. Akad. Wetensch. Proc. Ser. A, v. 65 = Indag. Math., v. 24, 1962, pp. 138148. MR. 25 #2690b. MR 0139255 (25:2690b)
 [14]
P. Wynn, "Converging factors for continued fractions. I,II," Numer. Math., v. 1, 1959, pp. 272320. MR 22 #6953. MR 0116158 (22:6953)
 [1]
 W. J. Cody, K. A. Paciorek & H. C. Thacher, "Chebyshev approximations for Dawson's integral," Math. Comp, v. 24, 1970, pp. 171178. MR 41 #2883. MR 0258236 (41:2883)
 [2]
 H. G. Dawson,"On the numerical value of ," Proc. London Math. Soc., v. 29, 1898, pp. 519522.
 [3]
 W. Fair, "Padé approximations to the solution of the Ricatti equation," Math. Comp., v. 18, 1964, pp. 627634. MR 29 #6630. MR 0169380 (29:6630)
 [4]
 D. G. Hummer, "Expansion of Dawson's function in a series of Chebyshev polynomials," Math. Comp., v. 18, 1964, pp. 317319. MR 29 #2967. MR 0165687 (29:2967)
 [5]
 E. Laguerre, "Sur le réduction en fractions continues d'une fraction qui satisfait à une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels," J. Math. Pures Appl., v. 1, 1885.
 [6]
 B. Lohmander, & S. Rittsten, "Table of the function ,'' Kungl. Fysiogr. Sällsk. i Lund Forh., v. 28, 1958, pp. 4552, MR 20 #1427. MR 0094919 (20:1427)
 [7]
 Y. Luke, "The Padé table and the method," J. Mathematical Phys., v. 37, 1958, pp. 110127. MR 20 #5558. MR 0099114 (20:5558)
 [8]
 J. H. McCabe, Continued Fraction Methods With Applications to First Order Ordinary Differential Equations, Ph D. Thesis, Brunel University, 1971.
 [9]
 J. H. McCabe, "An extension of the Padé table to include two point Padé approximations" (To appear.)
 [10]
 J. H. McCabe & J. A. Murphy, "Continued fraction expansions about two points" (To appear.)
 [11]
 J. A. Murphy, Rational Approximations to the Error Function, Internal project, Brunel University, 1966.
 [12]
 H. C. Thacher, Computation of the Complex Error Function by Continued Fractions, Blanch Anniversary volume, B. Mond (editor), Aerospace Research Laboratories, Washington, D. C.
 [13]
 P. Wynn, "The numerical efficiency of certain continued fraction expansions. IB," Nederl. Akad. Wetensch. Proc. Ser. A, v. 65 = Indag. Math., v. 24, 1962, pp. 138148. MR. 25 #2690b. MR 0139255 (25:2690b)
 [14]
 P. Wynn, "Converging factors for continued fractions. I,II," Numer. Math., v. 1, 1959, pp. 272320. MR 22 #6953. MR 0116158 (22:6953)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403710203
PII:
S 00255718(1974)03710203
Article copyright:
© Copyright 1974
American Mathematical Society
