Infinite Products of Holomorphic Functions

  • Reinhold Remmert
Part of the Graduate Texts in Mathematics book series (GTM, volume 172)

Abstract

Infinite products first appeared in 1579 in the work of F. Vieta (Opera, p. 400, Leyden, 1646); he gave the formula
$$\frac{2}{\pi } = \sqrt {\frac{1}{2}} \cdot \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2}} } \cdot \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2}} } } \cdot ...$$
for π (cf. [Z], p. 104 and p. 118). In 1655 J. Wallis discovered the famous product
$$\frac{\pi }{2} = \frac{{2\cdot 2}}{{1\cdot 3}}\cdot \frac{{4\cdot 4}}{{3\cdot 5}}\cdot \frac{{6\cdot 6}}{{5\cdot 7}}\cdot ...\cdot \frac{{2n\cdot 2n}}{{(2n - 1)\cdot (2n - 1)}}\cdot ...,$$
which appears in “Arithmetica infinitorum,” Opera I, p. 468 (cf. [Z], p. 104 and p. 119). But L. Euler was the first to work systematically with infinite products and to formulate important product expansions; cf. Chapter 9 of his Introductio. The first convergence criterion is due to Cauchy, Cours d’analyse, p. 562 ff. Infinite products had found their permanent place in analysis by 1854 at the latest, through Weierstrass ([Wei], p. 172 ff.).1

Keywords

Holomorphic Function Recursion Formula Partial Product Normal Convergence Infinite Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [B]
    Bernoulli, J.: Ars Conjectandi, Die Werke von Jakob Bernoulli, vol. 3, 107–286, Birkhäuser, 1975; German translation by R. HAUSSNER, Oswald’s Klassiker 107 (1895).Google Scholar
  2. [C]
    Von Fuss, P. J. (ed.): Correspondance mathématique et physique de quelques célèbres géomètres du XVIII ième siècle,St. Petersburg, 1843, 2 vols.; reprinted in 1968 by Johnson Reprint Corp.Google Scholar
  3. [D]
    Dickson, L. E.: History of the Theory of Numbers, vol. 2, Chelsea Publ.Co., New York, 1952.Google Scholar
  4. [Ei]
    Eisenstein, F. G. M.: Genaue Untersuchung der undendlichen Doppel-producte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenhängenden Doppelreihen, Journ. reine angew. Math. 35, 153–274 (1847); Math. Werke 1,357–478.Google Scholar
  5. [En]
    Eneström, G. E.: Jacob Bernoulli und die Jacobischen Thetafunction, Bibl. Math. 9, 3. Folge, 206–210 (1908–1909).Google Scholar
  6. [Eu]
    Leonhardi Euleri Opera omnia,sub auspiciis societatis scientarium natu-ralium Helveticae, Series I—IV A, 1911.Google Scholar
  7. [Ew]
    Ewell, J. A.: Consequences of Watson’s quintuple-product identity, Fi-bonacci Quarterly 20 (3), 256–262 (1982).MathSciNetMATHGoogle Scholar
  8. [Ga]
    Gauss, C. F.: Summatio quarundam serierum singularium, Werke 2, 9–45.Google Scholar
  9. [Go]
    Gordon, B.: Some identities in combinatorial analysis, The Quart. Journ.Math. (Oxford) 12, 285–290 (1961).MATHCrossRefGoogle Scholar
  10. [HW]
    Hardy, G. H. and E. M. Wright: An Introduction to the Theory of Numbers, 4th ed., Oxford, Clarendon Press, 1960.MATHGoogle Scholar
  11. I] Euler, L.: Introductio in Analysin Infinitorum,vol. 1, Lausanne 1748; inGoogle Scholar
  12. Eu], I-8; German translation, Einleitung in der Analysis des Unendlichen, 1885, published by Julius Springer; reprinted by Springer, 1983.Google Scholar
  13. [Jai]
    Jacobi, C. G. J.: Fundamenta nova theoriae functionum ellipticarum, Ges. Werke 1, 49–239.Google Scholar
  14. [Ja2]
    Jacobi, C. G. J.: Ueber unendliche Reihen, deren Exponenten zugleich in zwei verschiedenen quadratischen Formen enthalten sind, Journ. reine angew. Math. 37, 61–94 and 221–254 (1848); Ges. Werke 2, 217–288.Google Scholar
  15. [JF]
    Jacobi, C. G. J. and P. H. von Fuss: Briefwechsel zwischen C. G. J. Jacobi und P. H. von Fuss über die Herausgabe der Werke Leonhard Einers, ed. P. Stackel and W. Ahrens, Teubner, Leipzig, 1908.Google Scholar
  16. [JL]
    Jacobi, C. G. J.: Correspondance mathématique avec Legendre, Ges.Werke 1, 385–461.Google Scholar
  17. [Kn]
    Knopp, K.: Theory and Application of Infinite Series,Heffner Publishing Co., New York, 1947 (trans. R. C. H. YOUNG).Google Scholar
  18. [Kr]
    Kronecker, L: Theorie der einfachen und der vielfachen Integrale, ed. E. Netto, Teubner, Leipzig, 1894.MATHGoogle Scholar
  19. [M]
    Moore, E. H.: Concerning the definition by a system of functional properties of the function f(z) = (sin irz)/ir, Ann. Math. 9, 1st ser., 43–49 (1894).Google Scholar
  20. [N]
    Neher, E.: Jacobi’s Tripelprodukt Identität und ij-Identitäten in der Theorie affiner Lie-Algebren, Jber. DMV 87, 164–181 (1985).MathSciNetMATHGoogle Scholar
  21. [Nu]
    Numbers,Springer-Verlag, New York, 1993; ed. J. H. Ewing; trans. H. L.S. Orde.Google Scholar
  22. [P]
    Pringsheim, A.: Über die Convergenz unendlicher Produkte, Math. Ann. 33, 119–154 (1889).MathSciNetMATHGoogle Scholar
  23. [R]
    Ritt, J. F.: Representation of analytic functions as infinite products, Math. Zeitschr. 32, 1–3 (1930).MathSciNetMATHCrossRefGoogle Scholar
  24. [V]
    Valiron, G.: Théorie des fonctions, 2nd ed., Masson, Paris, 1948.MATHGoogle Scholar
  25. [Na]
    Watson, G. N.: Theorems stated by Ramanujan (VII): Theorems on con-tinued fractions, Journ. London Math. Soc. 4, 39–48 (1929).MATHCrossRefGoogle Scholar
  26. [Wei]
    Weierstrass, K.: Über die Theorie der analytischen Facultäten, Journ.reine angew. Math. 51, 1–60 (1856); Math. Werke 1, 153–221.Google Scholar
  27. [Weil]
    Weil, A.: Number Theory: An Approach through History, from Hammu-rapi to Legendre, Birkhäuser, 1984.Google Scholar
  28. [Z]
    Zeller„ C.: Zu Eulers Recursionsformel für die Divisorensummen, ActaMath. 4, 415–416 (1884).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Reinhold Remmert
    • 1
  1. 1.Mathematisches InstitutWestfälische Wilhelms—Universität MünsterMünsterGermany

Personalised recommendations