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Mathematics of Computation

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane

Authors: I. Gargantini and P. Henrici
Journal: Math. Comp. 21 (1967), 18-29
MSC: Primary 65.25; Secondary 41.00
MathSciNet review: 0240950
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Abstract: This report deals with the numerical evaluation of a class of functions of a complex variable that can be represented as Stieltjes transforms of nonnegative real functions. The considered class of functions contains, among others, the confluent hypergeometric functions of Whittaker and the Bessel functions. The method makes it possible, in principle, to compute the values of the function with an arbitrarily small error, using one and the same algorithm in whole complex plane cut along the negative real axis. Detailed numerical data are given for the application of the algorithm to the modified Bessel function $ {K_0}(z)$.

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  • [1] S. D. Conte & B. D. Fried, The Plasma Dispersion Function, Academic Press, New York, 1961. MR 0135916 (24:B1958)
  • [2] Handbook of Mathematical Functions, National Bureau of Standards Mathematics Series, No. 55, Government Printing Office, Washington, D. C., 1964.
  • [3] P. Henrici, "A subroutine for computations with rational numbers", J. Assoc. Comput. Mach., v. 3, 1956, pp. 6-9. MR 17, 673. MR 0074957 (17:673e)
  • [4] P. Henrici, Some Applications of the Quotient-Difference Algorithm, Proc. Sympos. Appl. Math. Vol. 15, Amer. Math. Soc., Providence, R. I., 1963, pp. 159-183. MR 28 #2632. MR 0159415 (28:2632)
  • [5] P. Henrici & P. Pfluger, "Truncation error estimate for Stieltjes fractions," (To appear.) MR 0212991 (35:3856)
  • [6] A. N. Khovanskii, The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, Noordhoff, Groningen, 1963. MR 27 #6058. MR 0156126 (27:6058)
  • [7] O. Perron, Die Lehre von den Keltenbrüchen. Vol. 2: Analytischfunklionenlheorelische Kettenbrüche, 3rd ed., Teubner, Stuttgart, 1957. MR 19, 25. MR 0085349 (19:25c)
  • [8] H. Rutishauser. "Der Quotienten-Differenzen-Algorithmus," Mitt. Inst. Angew. Math. Zürich, no. 7, MR 19, 686. MR 0089499 (19:686b)
  • [9] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge Univ. Press, Cambridge, 1962. MR 1349110 (96i:33010)

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Article copyright: © Copyright 1967 American Mathematical Society

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