Abstract
This paper contains a complete system of ALGOL procedures which enable arithmetic operations to be carried out upon complex numbers. Further procedures for carrying out the evaluation of certain elementary functions (e.g. ln, exp, sin, ...) of a complex variable are given. Application of these procedures is then illustrated by their use in the computation of the confluent hypergeometric function and the Weber parabolic cylinder function. Procedures relating to the application of the ε-algorithm to series of complex terms are described. Two integrated series of procedures, relating to Stieltjes typeS-fractions and to corresponding continued fractions respectively, are given. Complete programmes, which illustrate the use of these procedures, may be used for the computation of the incompleteβ-function, the incompleteΓ-function (of arguments of large and small modulus) and the Weber function.
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References
Backus, J. W. et al.,Report on the Algorithmic Language ALGOL 60, Num. Math., vol. 2, 1960, p. 106.
Fröberg, C. E.,Rational Chebyshev Approximation of Elementary Functions, BIT, vol. 1, 1961, p. 256.
Wynn, P.,On a Device for Computing the e m(Sn)Transformation, M.T.A.C., vo. 52, 1956, p. 663.
Szegö, G.,Über orthogonale Polynome, die zu einer gegebener Kurve der komplexen Ebene gehören, Math. Z., vol. 9, 1921, 0. 218.
Todd, J. and Warschawski, S. E.,On the Solution of the Lichtenstein-Gershgorin Integral Equation in Conformal Mapping: II Computational Experiments, N.B.S., Appl. Math. Ser. 42, p. 31.
Wynn, P.,The Rational Approximation of Functions which are Formally Defined by a Power Series Expansion, Maths. of Comp., vo. 14, 1960, p. 147.
Chebyshev, P.,Sur les Fractions Continues, Journ. de Math., vol. 8, 1858, p. 289.
Stieltjes, T. J.,Recherches sur les Fractions Continues, Annales de la Faculté des Sciences de Toulose, (1), 8, 1894, TI-122, 9, 1895, A5–57.
Markoff, A.,On Certain Applications of Algebraic Continued Fractions, Thesis, St. Petersburg, 1884.
Markoff, A.,Proof of the Convergence of Many Continued Fractions, Trans. Roy. Acad. Sci., St. Petersburg, 1893 (supp.).
Markoff, A.,On Functions Generated by Developing Power Series in Continued Fractions, Trans. Roy. Acad. Sci., St. Petersburg, 1894 (supp.).
Markoff, A.,Note sur les Fractions Continues, Bulletin de la Classe Physico-Mathématique de l'Académie Impériale des Sciences de Saint-Petersburg, vol. 5, 1895, p. 9.
Markoff, A.,Deux Demonstrations de la Convergence de Certaines Fractions Continues, Acta Mathematica, vol. 19, 1895, p. 93.
Markoff, A.,Nouvelles Applications des Fractions Continues, Memoires de l'Académie des Sciences de St. Petersburg, Classe Physico-Mathématique, vol. 3, 1896.
Markoff, A.,Nouvelles Applications des Fractions Continues, Math. Annalen, vol. 47, 1896, p. 579.
Shohat, J. A. and Tamarkin, J. D.,The Problem of Moments, Mathematical Surveys 1, Amer. Math. Soc. 1943.
Nevanlinna, R.,Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjes Momenten Problem, Annales Academiae Fenniae, (A), vol. 18, 1922.
Carleman, T.,Les Fractions Quasi-analytiques, Gauthier-Villars, Paris 1926.
Rutishauser, H.,Der Quotienten-Differenzen Algorithmus, Birkhauser Verlag, Basel, 1957.
Goodwin, E. T. and Staton, J., Table of\(\int_0^\infty {e^{ - u^2 } du/(u + x)} \), Quart. Journ. Mech. and Appl. Math., vol. 1, 1948, p. 319.
Perron, O.,Die Lehre von den Kettenbrüchen, vol. II, Teubner, Stuttgart, 1957.
van Vleck, E. V.,On the Convergence of Algebraic Continued Fractions whose Coefficients have Limiting Values, Trans. Amer. Math. Soc., vol. 5, 1904, p. 253.
Ramanujan, S.,Collected Papers, Cambridge 1927.
Wynn, P.,The numerical Efficiency of Certain Continued Fraction Expansions, Proc. Kon. Ned. Akad. Wetensch. Amsterdam, vol. 65, ser. A, 1962, p. 127.
Wynn, P.,Numerical Efficiency Profile Functions, Proc. Kon. Ned. Akad. Wetensch. Amsterdam, vol. 65, ser. A, 1962, p. 118.
Erdélyi, A. et al.,Higher Transcendental Functions, McGraw-Hill.
Clenshaw, C. W.,Chebyshev Series for Mathematical Functions, Mathematical Tables, vol. 5, H.M.S.O., London 1962.
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Communication MR 51 of the Computation Department of the Mathematical Centre, Amsterdam.
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Wynn, P. An arsenal of ALGOL procedures for complex arithmetic. BIT 2, 232–255 (1962). https://doi.org/10.1007/BF01940171
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DOI: https://doi.org/10.1007/BF01940171