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A rational approximation to Weierstrass' $ P$-function


Author: Ulrich Eckhardt
Journal: Math. Comp. 30 (1976), 818-826
MSC: Primary 65D20; Secondary 33A25
DOI: https://doi.org/10.1090/S0025-5718-1976-0421042-0
MathSciNet review: 0421042
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Abstract: A rational approximation to Weierstrass' $ \wp$-function in the equianharmonic case for unit period parallelogram is given. With a third-degree numerator polynomial and a fourth-degree denominator polynomial the maximal error for $ \vert z\vert < 1/\sqrt 3 $ becomes $ 3 \cdot {10^{ - 14}}$.

The approximation of $ \wp(z)$ is then used to calculate a rational approximation to $ \wp'(z)$ together with an error bound.


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  • [1] M. ABRAMOWITZ & I. A. STEGUN, Editors, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, vol. 55, U.S. Government Printing Office, 1964; reprint, Dover, New York, 1966. MR 34 #8606.
  • [2] N. I. AHIEZER, Elements of the Theory of Elliptic Functions, GITTL, Moscow, 1948; 2nd rev. ed., "Nauka", Moscow, 1970. (Russian) MR 12, 409; 44 #5517. MR 0288319 (44:5517)
  • [3] H. T. DAVIS, Tables of the Mathematical Functions, vol. II, Principia Press of Trinity University, San Antonio, Texas, 1963. MR 0158099 (28:1325b)
  • [4] U. ECKHARDT, Zur Berechnung der Weierstrassschen Zeta- und Sigma-Funktion, Berichte der KFA Jülich, Jül-964-MA, June 1973.
  • [5] O. EMERSLEBEN, "Erweiterung des Konvergenzbereichs einer Potenzreihe durch Herausnahme von Singularitäten, insbesondere zur Berechnung einer Zetafunktion zweiter Ordnung," Math. Nachr., v. 31, 1966, pp. 195-220. MR 33 #4022. MR 0195824 (33:4022)
  • [6] E. GRAESER, Einführung in die Theorie der elliptischen Funktionen und deren Anwendungen, Verlag von R. Oldenbourg, München, 1950. MR 12, 607. MR 0039848 (12:607e)
  • [7] A. G. GREENHILL, The Application of Elliptic Functions, Dover, New York, 1959. MR 22 #2724. MR 0111864 (22:2724)
  • [8] W. MAGNUS & F. OBERHETTINGER, Anwendung der elliptischen Funktionen in Physik und Technik, Die Grundlehren der math. Wissenschaften, Band 55, Springer-Verlag, Berlin, 1949. MR 11, 104. MR 0031129 (11:104f)
  • [9] T. H. SOUTHARD, "Approximation and table of the Weierstrass $ \wp$ function in the equianharmonic case for real argument," MTAC, v. 11, 1957, pp. 99-100. MR 19, 182. MR 0086416 (19:182c)
  • [10] T. H. SOUTHARD, "Weierstrass elliptic and related functions," in [1, Chapter 18].
  • [11] V. K. TKAČENKO, "On vortex lattices," Ž. Èksper. Teoret. Fiz., v. 49, 1965, pp. 1875-1883. (Russian)
  • [12] F. TÖLKE, Praktische Funktionenlehre. Band II: Theta-Funktionen und spezielle Weierstrasssche Funktionen, Springer-Verlag, Berlin and New York, 1966. MR 35 #3089.
  • [13] F. TRICOMI, Elliptische Funktionen, Transl. and edited by M. Krafft, Mathematik und ihre Anwendungen in Physik and Technik, Reihe A, Band 20, Akademische Verlagsgesellschaft, Geest & Portig K.-G., Leipzig, 1948. MR 10, 532. MR 0029001 (10:532c)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1976-0421042-0
Article copyright: © Copyright 1976 American Mathematical Society

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