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Chapter 8 Advanced Topics in the Theory of Functions

verfasst von : Umberto Bottazzini, Jeremy Gray

Erschienen in: Hidden Harmony—Geometric Fantasies

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Abstract

By 1880 the main features of the present-day complex function theory had been created, although not assembled into their modern order. Cauchy and Riemann were dead, but Weierstrass was in firm control of his own version of the theory and lecturing on it as part of his 2-year cycle of lectures in Berlin. As we saw in the previous chapter, complex function theory in its various forms had found in widening number of successful applications to other domains of both pure and applied mathematics. In this chapter we look at how the ideas of the founders led to important developments in the theory of complex functions itself and to the discovery of some of the central features of complex function theory as a rich research topic in its own right.

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Vorheriges Kapitel Chapter 7 Complex Function Theory and Differential Equations

Nächstes Kapitel Chapter 9 Several Complex Variables

Edwards (1984); Lützen (1990); Nicholson (1993); and Neumann (2007).

Klein (1879) showed that the Galois group of the modular equation was much smaller than the Galois group of the general polynomial equation of degree 7, but the reduction process would solve any 7th degree equation whose Galois group was that of the modular equation, thus explaining Kronecker’s result.

Here suppressed, see Jordan (1870) and, for a modern treatment, Umemura, in (Mumford, 1984, 3.261–3.272).

See Thomae (1870a); Lindemann (1884) and Lindemann (1892).

Discussed at length in (Gray 2000a, Chap. 4).

See Gray (2000a).

Between 1877 and 1878 Klein and Brioschi corresponded intensively on the solution of the quintic equation and related matters. Klein’s letters to Brioschi are kept in the library of the Politecnico in Milan.

The following year, Picard published an extended version of this material in his (1880).

Picard missed the trick of using Liouville’s theorem and proved a similar theorem about harmonic functions instead.

Picard, Traité d’analyse 2, 122.

Hilbert’s 21st Paris problem asks: given a quotient group of a fundamental group of a punctured Riemann surface, is there a differential equation having its singular points only at the punctures and the given group as its monodromy group? Although it was answered affirmatively by G.D. Birkhoff in his (1913a) and by Plemelj (1908b) and their work stood for over 70 years, it was shown to be flawed by Bolibruch. See Anosov and Bolibruch (1994) and for an indication of the history, (Gray 2000b, 139–140).

First published as (Poincaré 1881a,b,c), see Gray (2000a).

See Poincaré, Oeuvres 11, 13–25.

He knew of Fuchs’s work because Fuchs was a friend of Hermite.

First published as (Poincaré 1997).

See (Poincaré 1908b, 52). He still did not win; the prize went to Halphen for a memoir on differential equations and invariant theory (Halphen 1884).

See Gray (2000a) for an account and Poincaré (1885a) for an English translation of Poincaré’s major papers (1882c,d,e) and an interesting commentary by J. Stillwell.

See, e.g., Klein, Ges. Math. Abh. 3, 587–621 and Poincaré, Oeuvres 11, 26–65.

Frobenius was a late developer.

Mittag-Leffler raised the subject of Schwarz’s feelings in a letter he wrote to Poincaré on 18 July 1882, where he observed that although Fuchs was full of admiration for Poincaré’s “beautiful discoveries” Schwarz was almost “suffocating with anger”. Poincaré replied on 27 July 1882 that he had re-read the relevant papers and concluded that he could do nothing to calm Schwarz down, because Schwarz was really angry with himself “for having had an important result in his hands and not profiting from it. And I can do nothing about that”. As for the choice of name, Poincaré felt he had nothing to add. See Nabonnand (1999), 100–101.

Fuchsian and Kleinian functions are collectively called automorphic functions.

In (Poincaré 1884a, 332).

And perhaps little understood. For example, Mittag-Leffler wrote to Hermite on 13 October 1884 that “I have not understood Poincaré at all for a very long time and he really expresses himself in a very obscure manner”. Quoted in (Dugac 1984, 203).

His Times obituary dwelled on this point to the exclusion of mathematics.

The first had already appeared in Picard (1883). He returned again to this theorem in Picard (1912).

Such as those where a condition on the derivative is specified, or where it is not a potential function that is involved, but, say, the minimal surface equation. See Gray (2000b).

This conception of a continuum Hurwitz had learned from Weierstrass in 1878, see Sect. 6.5.3 above.

In later terminology, this asks about what sets can be a domain of holomorphy.

For a survey of complex function theory as presented at the ICM’s up to 1932, see (Bottazzini and Gray 1996).

See Parshall and Rowe (1994) for details.

See (Jordan 1882–1887, 3, 587–594). In the 2nd ed. of the Cours (1893–1896) this note, essentially unchanged, was moved to vol. 1, pp. 92–100.

This refers to Courant (1912).

It helps to think of this slit as going to infinity.

Bieberbach’s membership of the Nazi Party later made him notorious.

See Montel (1907) and Montel (1910). Montel’s work is discussed below, see Sect. 8.6.

We discuss this theorem in Sect. 8.3.2.1.

This recalls Riemann’s proof of the Riemann mapping theorem, Sect. 5.2.4.

In this way Poincaré took care of the arbitrariness introduces by the sweeping out.

Poincaré also took care of the way sweeping out had to be used, which complicates the definition of the limiting process.

The memoir also analysed the cases when a given simply connected region is to be represented conformally not on a circle but on the whole plane and showed how these cases could be distinguished. In modern terms, he dealt with the case where the simply connected covering space is not the disc but the plane. This is the famous recognition problem later much discussed by Ahlfors and others.

See Gray (2000b) for the references cited there.

Presumably analytic, but Koebe did not say.

See (Klein Ges. Math. Abh. 3, 731–741) and (van Dalen 1999–2005, 180–193).

Fubini (1908) is likewise critical.

For an account of de Branges’ work and the eventual resolution of Bieberbach’s conjecture, see the eye-witness account (Fomenko and Kuz’mina 1986).

A function that is harmonic away from finitely many points where it is infinite like logr and which vanishes on the boundary R of the domain B.

After graduating from the University of Berlin, Schottky taught there as Privatdozent until 1882 when he was appointed to a chair at the Eidgenössische Polytechnische Schule in Zürich. Ten years later he moved to the University of Marburg, and after ten more years he returned to his home University in Berlin where he stayed until his retirement in 1922. His work dealt mainly with complex function theory and cannot be better summarised than Freudenthal (1975b, 213) did in his biography in the DSB:“His work is difficult to read. Although he was a student of Weierstrass, his approach to function theory was Riemannian in spirit, combined with Weierstrassian rigor”.

In fact, as he showed, one can take u(z) = p(z).

We omit Schottky’s account of the exceptional points. They arise from the fixed points of the transformations γ ′ _j.

For an accessible modern account, see Hejhal (1972).

He later developed these ideas at greater length in his book (Osgood 1907 and subsequent editions).

See the essay by Remmert in (Weyl 1997), who rightly notes the remarkable influence of Weyl’s book.

An observation Poincaré had made in his (1907a).

Triangulation of a Riemann surface was only established properly in (Radó 1925).

He repeated this argument “essentially unchanged” in his (1955, 95).

In his (1955, 131) he wrote that he regarded this “as an essential improvement of the method”.

The Weierstrass gap theorem was recalled only in the 1955 edition (p. 140).

Quoted in (Maz’ya and Shaposhnikova 1998, 39).

The modern term is genus, but we have kept the French to avoid confusion with the concept of the genus of a Riemann surface.

See (Maz’ya and Shaposhnikova 1998, 114) which is the definitive biography of Hadamard.

Readers may at least consult (Collingwood 1959; Fréchet 1965; May 1970).

The proof was corrected in (Khavinson and Shapiro 1994).

Among the functions that behave very like a polynomial is Riemann’s ξ function.

See (Maz’ya and Shaposhnikova 1998, 306). Others working this area independently were Fabry and Lindelöf.

Hadamard’s series are lacunary series because n _k > k − 1.

Hadamard explained that when λ is not an integer the function is of genre E, where E + 1 is the least integer greater than λ, but if λ is the integer E + 1, the genre may be either E or E + 1.

This paper was actually printed on February 24, 1898, which is how Borel was able to refer to it.

“Logical” was the French term for formal and non-constructive.

On Nevanlinna’s life, and for an introduction to his work, see the biography (Lehto 2008).

We are grateful for Ivor Grattan-Guinness for drawing our attention to this manuscript, and to the Archivist of the University of Liverpool for supplying us with a copy.

The result was announced in Comptes rendus 141 (1905) 305–307. In the paper containing the fallacious proof, Bull SMF 34 (1906) 30–39, Boutroux acknowledged Schottky’s priority.

He was to give a streamlined version of the paper in his (1917).

In this paper, Bohr and Landau showed that the zeta function ζ(s) is unbounded in the region where Re(s) > 1, and applied Schottky’s theorem to deduce several delicate properties of the zeta function.

For an account, see (Segal 1981, Chap. II) or (Landau 1929).

See (Mittag-Leffler 1909, 75).

A minor consequence was that a further Swedish Royal Prize, this one of 3,000 kr for a paper on the theory of analytic functions, was abandoned, see Acta mathematica 37 (1914).

Our thanks to Bob Burckel and Alan Beardon for their comments on a previous version of this material, (Gray 2000c). See also Bottazzini’s The Higher Calculus, 179–180.

See Pringsheim (1929).

This is the 3rd edn. of Bieberbach (1921b).

Kamke quoted from the 4th edn. of Knopp (1913), which reproduced the proof given in the first 1913 edition at the same pages.

For Runge’s biography, see Richenhagen (1985).

Runge may have already been departing from Weierstrassian orthodoxy. To Sonya Kovalevskaya Runge confided that “The subject does not interest me enough any more. These developments are too general. What general developments, when the function is above all only defined in any way whatever, and indeed it is defined by the Cauchy integral in a very lovely way. And whether there are functions with arbitrary natural boundaries is also a matter of some indifference to me”. Runge to Kovalevskaya, 24 January 1884, quoted in (Richenhagen 1985, 62).

This ability to vary the location of the poles of a rational function (within certain domains) is called “pole pushing” in the modern literature. The modern form of this result isolates the poles of the rational function R(x) and considers a holomorphic function on a closed bounded but not necessarily simply connected domain B; the claim is that the given function can be approximated uniformly by a sequence of rational functions having poles only in the components of the complement of B in \(\mathbb{C}\).

Montel observed in his (1912b) that uniform convergence on the boundary is a necessary condition for term-by-term differentiation to be valid.

See (Painlevé 1898). Similar results had been obtained in (Hilbert 1897a).

Milton Brockett Porter was a student of Halsted’s at the University of Texas. He took his PhD at Harvard in 1897 and returned to Texas as Halsted’s successor in 1903, where he remained for the next 42 years. See Parker (2005).

As was customary, his thesis, or a modified version of it, was then published in the Annales of the École Normale (1907). In later life he was Dean of Faculty of Science during the Second World War; Dieudonné (1990, 649) wrote that “he was able to uphold the honour of the French university” in those difficult times.

See (Siegmund–Schultze 1998 and 2003) and (Fraser 2003).

See (Montel 1912a), expanded as (Montel 1912b).

The Schottky inequality principle re-appeared in the French literature in 1918 in the third edition of Goursat’s Cours d’analyse, vol. 2, Appendix, pp. 651–663, which followed the lines of (Schottky 1906).

See the book (Alexander et al. 2011) for a thorough history documenting the subject’s origins in work on functional equations and its continuation to the celebrated prize competition and beyond, and also (Alexander 1994) and (Alexander 1996).

Fatou (1917a, b) showed there is always such a fixed point.

As mentioned by Audin, Lattès died of typhoid in the summer of 1918 at the age of 45. His work is discussed in (Alexander et al. 2011, Chap. 7).

A fixed point z of a function f is said to be repelling if | f ′(z) | > 1. A point of period p is repelling if | (f ^p)′(z) | > 1. Similarly, fixed and periodic points for which the absolute value of the derivative is less than 1 are called attractive, and neutral if that absolute value = 1.

A set is perfect if it coincides with its derived set.

This problem became known as the centre problem and required techniques from number theory for its further treatment.

Fatou denoted the function R.

See (Beardon 1991, 50), the reference is to (Blanchard 1984).

As Alexander (1994, 114) has also noted.

100

Fatou (1920, 40), quoted in (Alexander 1994, 132).

101

See Beardon (1991) for a good starting point and overview.

102

Geometry could not be kept away indefinitely, as Ahlfors would demonstrate in the 1930s.

Alexander, D.S. 1994. A history of complex dynamics from Schröder to Fatou and Julia. Vieweg & Sohn, Braunschweig.MATH

Alexander, D.S. 1996. An episodic history of complex dynamics from Schröder to Fatou and Julia. Suppl. Rend. Palermo (2) 44, 57–83.

Alexander, D.S., F. Iavernaro, and A. Rosa. 2011. Early days in complex analysis. HMath 38. Providence, RI.

Anosov, D.V. and A.A. Bolibruch. 1994. The Riemann–Hilbert problem. Vieweg & Sohn, Braunschweig.MATH

Audin, M. 2009. Fatou, Julia, Montel. Le grand prix des sciences mathématiques de 1918, et après …. Springer, New York. Engl. trl. as Fatou, Julia, Montel. The great prize of the mathematical sciences of 1918, and beyond Springer, New York. 2011.

Baker, H.F. 1895. Abelian functions. CUP, Cambridge.

Beardon, A.F. 1991. The iteration of rational functions. Springer, New York.CrossRef

Beltrami, E. 1868a. Saggio di interpretazione della geometria non–euclidea. G. di Mat. 6, 285–315 in Op. Mat. 1, 374–405.

Bernays, P. 1913. Zur elementaren Theorie der Landauschen Funktion \(\varphi (a)\). Natur. Gesell. Zürich 58, 203–238.

Bieberbach, L. 1914. Zur Theorie und Praxis der konformen Abbildung. Rend. Palermo 38, 98–112.MATHCrossRef

Bieberbach, L. 1915. Einführung in die konforme Abbildung. G. J. Göschen, Berlin.MATH

Bieberbach, L. 1916. Über ein Extremalproblem im Gebiet der konformen Abbildung. Math. Ann. 77, 153–172.MathSciNetMATHCrossRef

Bieberbach, L. 1921b. Lehrbuch der Funktionentheorie. vol. 1. Teubner, Leipzig und Berlin. 3rd ed. Teubner, Leipzig und Berlin 1930. 4th ed. Teubner, Leipzig und Berlin 1934. Rep. Chelsea, New York 1945. Johnson Repr. Corp., New York 1988.

Blake, E. M. 1894. Bibliography of mathematical dissertations. Bulletin of the New York Mathematical Society 3, 125–127.MathSciNetCrossRef

Blanchard, P. 1984. Complex analytic dynamics on the Riemann sphere. Bull. AMS (2) 11, 85–141.

Bölling, R. (ed.). 1993. Briefwechsel zwischen Karl Weierstrass und Sofja Kowalewskaja. Akademie–Verlag, Berlin.MATH

Borel, E. 1897. Sur les zeros des fonctions entières. Acta 20, 357–396 in Oeuvres 1, 577–616.

Borel, E. 1900. Leçons sur les fonctions entières. Gauthier–Villars, Paris.MATH

Borel, E. 1901a. Remarques relatives à la communication de M. Mittag-Leffler. Comptes rendus du Congrès international des Mathématiciens, Paris 1900, 277–278. Gauthier–Villars, Paris in Oeuvres 2, 705–706.

Borel, E. 1901b. Leçons sur les séries divergentes. Gauthier–Villars, Paris.

Borel, E. 1917. Leçons sur les fonctions monogènes uniformes d’une variable complexe. Gauthier-Villars, Paris.MATH

Bottazzini, U. and J.J. Gray. 1996. Complex function theory from Zurich (1897) to Zurich (1932). Suppl. Rend. Palermo (2) 44, 85–111.

Boutroux, P. 1905. Sur les fonctions entières d’ordre entier. Verhandlungen des dritten internationalen Mathematiker–Kongresses, Heidelberg 1904, 253–257. Teubner, Leipzig.

Brodén, T. 1905. Bemerkungen über die Uniformisierung analytischer Funktionen. Berling, Lund.

Brouwer, L.E.J. 1910. Beweis des Jordanschen Kurvensatzes. Math. Ann. 69, 169–175 in CW 2, 377–383.

Brouwer, L.E.J. 1912b. Über den Kontinuitätsbeweis für das Fundamentaltheorem der automorphen Funktionen im Grenzkreisfalle. JDMV 21, 154–157 in CW 2, 568–571.

Burnside, W. 1891. On a class of automorphic functions. Proc. LMS 23, 49–88 in Coll. Papers 1, 249–288.

Burnside, W. 1892. Further note on automorphic functions. Proc. LMS 23, 281–295 in Coll. Papers 1, 321–335.

Collingwood, E.F. 1959. Émile Borel. J LMS 34, 488–512. Addendum: Émile Borel. J LMS 35 (1960), 384.

Courant, R. 1912. Über die Anwendung des Dirichlets Prinzipes auf die Probleme der konformen Abbildung. Math. Ann. 71, 145–183.MathSciNetMATHCrossRef

Darboux, G. 1878. Mémoire sur l’approximation des fonctions de très–grands nombres, et sur une classe étendue de développements en série. J de math. (3) 4, 5–56; 377–416.

Darboux, G. 1909. Les origines, les méthodes et les problèmes de la géométrie infinitésimale. Atti del IV Congresso Internazionale dei Matematici, Roma 1908 Castelnuovo, G. (ed.). 1, 105–122. Tipografia Accademia dei Lincei, Roma.

Dedekind, R. 1877. Schreiben an Herrn Borchardt über die Theorie der elliptische Modulfunctionen. JfM 83, 265–292 in Ges. Math. Werke 1, 174–201.

Dehn, M. and P. Heegard. 1907. Analysis situs. EMW III A B 3, 153–220.

Dienes, P. 1913. Leçons sur les singularités des fonctions analytiques; professées à l’université de Budapest. Gauthier-Villars, Paris.MATH

Dienes, P. 1931. Taylor series. An introduction to the theory of functions of a complex variable. Clarendon Press, Oxford. Rep. Dover, New York 1957.

Dieudonné, J. 1966. L’œuvre scientifique de Paul Montel. In Paul Montel, mathématicien niçois. Essais et témoignages, 85–90. Ville de Nice.

Dieudonné, J. 1990. Montel. DSB 18, Suppl. 2, 649–650.

Dugac, P. (ed.). 1984. Lettres de Charles Hermite à Mittag-Leffler (1884–1883). Cahiers du Séminaire d’Histoire des Mathématiques 5, 49–285.

Edwards, H.M. 1984. Galois theory. Springer, New York.MATH

Eisenstein, G. 1847a. Genaue Untersuchung der unendliche Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind. JfM 35, 153–274 in Math. Abh. 213–334; in Math. Werke 1, 357–478.

Fatou, P. 1906. Séries trigonométriques et séries de Taylor. Acta 30, 335–400.MathSciNetMATH

Fatou, P. 1919. Sur les équations fonctionnelles. Bull. SMF 47, 161–271.MathSciNetMATH

Fatou, P. 1920. Sur les équations fonctionnelles. Bull. SMF 48, 33–94; 208–314.MathSciNet

Fomenko, O.M. and Kuz’mina, G.V. 1986. The last 100 days of the Bieberbach conjecture. Mathematical Intelligencer 8, 40–47. Rep. in (Gray and Wilson 2000, 429–441).

Fraser, C. 2003. The calculus of variations: a historical survey. In (Jahnke 2003, 355–383).

Fréchet, M. 1906. Sur quelques points du calcul fonctionnel. Rend. Palermo 22, 1–74.MATHCrossRef

Fréchet, M. 1965. La vie et l’oeuvre d’Émile Borel. Enseignement mathématique (2) 11, 1–9.

Freudenthal, H. 1975b. Schottky, Friedrich Hermann. DSB 12, 212–213.

Fubini, G. 1908. Introduzione alla teoria dei gruppi discontinui e delle funzioni automorphe. Spoerri, Pisa.

Gispert, H. 1987. La correspondance de G. Darboux avec J. Houël: chronique d’un rédacteur (déc. 1869 – nov. 1871). Cahiers du Séminaire d’Histoire des Mathématiques 8, 67–202.

Gispert. H. 1991. La France mathématique. Cahiers d’histoire et de philosophie des sciences 34.

Goursat, E. 1900. Sur la définition générale des fonctions analytiques, d’après Cauchy. Trans. AMS 1, 14–16.MathSciNetMATH

Gray, J.J. 1994. On the history of the Riemann mapping theorem. Suppl. Rend. Palermo (2) 34, 47–94.

Gray, J.J. 2000a. Linear differential equations and group theory from Riemann to Poincaré. Birkhäuser, Boston and Basel.MATH

Gray, J.J. 2000b. The Hilbert challenge. OUP, Oxford. French trl. as Le défi de Hilbert. Un siècle de mathématiques. Dunod, Paris 2003.

Gray, J.J. 2000c. Goursat, Pringsheim, Walsh, and the Cauchy integral theorem. Mathematical Intelligencer 22, 60–66; 77.

Hadamard, J. 1892. Essai sur l’étude des functions donnés par leur developpement de Taylor. J de math. (4) 8, 101–186 in Oeuvres 1, 792–877.

Hadamard, J. 1893. Étude sur les propriétés des fonctions entières et en particulier d’une fonction considerée par Riemann (Mémoire couronné par l’Académie, Grand Prix des sciences mathématiques). J de math. (4) 9, 171–215 in Oeuvres 1, 103–147.

Hadamard, J. 1897. Théorème sur les séries entières. CR 124, 492. [Not in Oeuvres].

Hadamard, J. 1899. Théorème sur les séries entières. Acta 22, 55–64 in Oeuvres 1, 93–101.

Hadamard, J. 1901. La série de Taylor et son prolongement analytique. C. Naud, Paris. [Not in Oeuvres].

Halphen G.H. 1884. Sur le reduction des équations différentielles linéaires aux formes intégrables. Mémoires presentés par divers savants 28, 1–260 in Oeuvres 3, 1–260.

Harnack, A. 1887. Grundlagen der Theorie des logarithmischen Potentiales [etc]. Teubner, Leipzig.

Heffter, L. 1902. Reelle Curvenintegration. Göttingen Nachr. 26–52.

Hejhal, D. 1972. Theta functions, kernel functions, and Abelian integrals. Memoirs Series AMS, 129. AMS, Providence, RI.

Hensel, K. and G. Lansdberg. 1902. Theorie der algebraischen Funktionen einer Variabeln und ihre Anwendung auf algebraische Kurven und Integrale. Teubner, Leipzig. Rep. Chelsea, New York 1965.

Hilbert, D. 1891. Ueber die stetige Abbildung einer Linie auf ein Flächenstück. Math. Ann. 38, 459–460 in Ges. Abh. 3, 1–2.

Hilbert, D. 1897a. Die Theorie der algebraischen Zahlkörper. JDMV 4, 175–546 in Ges. Abh. 1, 63–363. Engl. trl. as The theory of algebraic number fields. Lemmermeyer, F. and N. Schappacher (eds). Springer, New York 1998.

Hille, E. 1962. Analytic function theory. 2 vols. Ginn & Co., Boston.

Hurwitz, A. 1881. Grundlagen einer independenten Theorie der elliptischen Modulfunktionen [etc]. Math. Ann. 18, 528–592 in Math. Werke 1, 1–66.

Hurwitz, A. 1883. Beweis des Satzes, dass eine einwertige Funktion beliebig vieler Variabeln [etc]. JfM 95, 201–206 in Math. Werke 1, 147–152.

Hurwitz, A. 1888. Ueber diejenigen algebraischen Gebilde, welche eindeutige Transformationen in sich zulassen. Math. Ann. 32, 290–308, in Math. Werke 1, 241–259.

Hurwitz, A. 1897. Über die Entwicklung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit. Verhandlungen der ersten internationalen Mathematiker–Kongresses, 91–112. Teubner, Leipzig in Math. Werke 1, 461–480.

Johansson, S. 1905. Über die Uniformisierung Riemannscher Flächen mit endlicher Anzahl Windungspunkte. Acta Soc. Sci. Fennicae 33, Nr. 7.

Johansson, S. 1906a. Ein Satz über die konformen Abbildung einfach zusammenhängender Riemannscher Flächen auf den Einheitskreis. Math. Ann. 62, 177–183.MathSciNetMATHCrossRef

Jordan, C. 1870. Traité des substitutions et des équations algébriques. Gauthier–Villars, Paris. Rep. Gauthier–Villars, Paris 1957. Rep. Gabay, Paris 1989.

Jordan C. 1878. Mémoire sur les équations différentielles linéaires à intégrale algébrique. JfM 84, 89–215 in Oeuvres 2, 13–140.

Jordan, C. 1882–1887. Cours d’analyse de l’École Polytechnique. 3 vols. Gauthier–Villars, Paris. 2nd ed. Gauthier–Villars, Paris 1893–1896. 3rd ed. Gauthier–Villars, Paris 1909–1915. Rep. Gauthier–Villars, Paris 1982–1987. Rep. Gabay, Paris 1991.

Julia, G. 1917. Etude sur les formes binaires non quadratiques à indéterminées réelles ou complexes, ou à indéterminées conjuguées. Mém. Acad. Sci. Paris 55, 1–296.

Julia, G. 1918. Mémoire sur l’iteration des fonctions rationelles. (Grand Prix des Sciences Mathématiques 1818). J de math. (8) 1, 47–245 in Oeuvres 1, 121–321.

Julia, G. 1933. Essai sur le développment de la théorie des fonctions de variables complexes. Gauthiers–Villars, Paris in Congrès international des Mathématiciens, Zürich 1932, 1, 102–127 in Oeuvres 3, 372–429.

Khavinson, D. and H.S. Shapiro. 1994. The heat equation and analytic continuation: Ivar Fredholm’s First Paper. Expositiones mathematicae 12, 79–95.MathSciNetMATH

Klein, C.F. 1879. Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Math Ann. 14, 428–471, in Ges Mat. Abh., 3, 90–136.

Klein, F. 1882a. Ueber Riemanns Theorie der algebraischen Funktionen und ihrer Integrale. Teubner, Leipzig in Ges. Math. Abh. 3, 499–573. Engl. trl. as On Riemann’s theory of algebraic functions and their integrals. Macmillan and Bowes, Cambridge. Rep. Dover, New York 1963.

Klein, F. 1882b. Über eindeutige Funktionen mit linearen Transformationen in sich. Math. Ann. 19, 565–568 in Ges. Math. Abh. 3, 622–626.

Klein, C.F. 1890a. Zur Theorie der Abel’schen Funktionen. Math. Ann. 36, 1–83 in Ges. Math. Abh. 3, 388–474.

Klein, C.F. 1921–1923. Gesammelte mathematische Abhandlungen. Fricke, R., Ostrowski, A. M., Vermeil, H. and E. Bessel–Hagen (eds). 3 vols. Springer, Berlin.

Knopp, K. 1913. Funktionentheorie. 2 vols. G. J. Göschen’sche Verlagshandlung, Berlin und Leipzig. Many subs editions. Engl. trl. Theory of functions. 2 vols. Dover, New York 1945–1947. Rep. Dover New York 1996.

Koebe, P. 1907a. Über die Uniformisierung reeller algebraischer Kurven. Göttingen Nachr. 177–190.

Koebe, P. 1907b. Über die Uniformisierung beliebiger analytischer Kurven. Göttingen Nachr. 191–210.

Koebe, P. 1907c. Über die Uniformisierung beliebiger analytischer Kurven (Zweite Mitteilung). Göttingen Nachr. 633–669.

Koebe, P. 1908. Über die Uniformisierung beliebiger analytischer Kurven (Dritte Mitteilung). Göttingen Nachr. 337–358.

Koebe, P. 1909c. Über die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginärer Substitutionsgruppe. Göttingen Nachr. 68–76.

Koebe, P. 1909d. Ueber ein allgemeines Uniformisierungsprinzip. Atti del IV Congresso Internazionale dei Matematici, Roma 1908 Castelnuovo, G. (ed.). 2, 25–30.

Koebe, P. 1910a. Über die Uniformisierung der algebraischen Kurven, II. Math. Ann. 69, 1–81.MathSciNetMATHCrossRef

Koebe, P. 1910b. Über die Uniformisierung beliebiger analytischer Kurven, I. JfM 138, 192–253.MATH

Koebe, P. 1910c. Über die Hilbertsche Uniformisierungsmethode. Göttingen Nachr. 59–74.

Koebe, P. 1912a. Ueber eine neue Methode der konformen Abbildung und Uniformisierung. Göttingen Nachr. 844–848.

Koebe, P. 1912b. Referat über automorphe Funktionen und Uniformisierung. JDMV 21, 157–163.

Koenigs, G. 1884. Recherches sur les intégrales de certaines équations fonctionnelles. Annales ENS (3) 1, Suppl. 3–41.

Landau, E. 1906. Über den Picardschen Satz. Natur. Gesell. Zürich 51, 252–318.

Leau, L. 1899. Recherches des singularités d’une fonction définie par un développement de Taylor. J. de math. (5) 5, 365–425.

Lebesgue, H. 1902. Intégrale, longueur, aire. Ann. di Mat. (3) 7, 231–359 in Oeuvres scientifiques 1, 203–331.

Lebesgue, H. 1906 Leçons sur les séries trigonométriques professées au Collège de France. Gauthier-Villars, Paris. [Not in Oeuvres scientifiques].

Lehto, O. 2008. Erhabene Welten: das Leben Rolf Nevannlinas, Birkhäuser, Basel.

Le Roy, É. 1900. Sur les séries divergentes et les fonctions définies par un développement de Taylor. Annales de la Faculté des sciences de Toulouse (2) 2, 317–430.

Lévy, P. 1912. Sur une généralisation des théorèmes de MM. Picard, Landau et Schottky. CR 153, 658–660.

Lindelöf, E. 1908. Sur une extension d’un principe classique de l’analyse et sur quelques propriétés de fonctions monogènes dans le voisinage d’un point singulier. Acta 31, 381–406.MATH

Lindelöf, E. 1912. Sur le théorème de M. Picard. Compte rendu du Congrès des mathématiciens scandinaves tenu à Stockholm, 112–136.

Lindelöf, E. 1915. Sur un principe geénéral de l’analyse à la théorie de la représentation conforme. Imprimerie de la Société de littérature finnoise, Helsingfors.

Lindemann, F. 1884. Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen, I. Göttingen Nachr. 245–248.

Lindemann, F. 1892. Ueber die Auflösung algebraischer Gleichungen durch transcendente Functionen, II. Göttingen Nachr. 292–298.

Lindemann, F. 1899. Zur Theorie der automorphen Functionen. Sitz. München 29, 423–454.

Lützen, J. 1990. Joseph Liouville, 1809–1882. Master of pure and applied mathematics. Springer, New York.MATH

Malmsten, C.J. 1865. Om definita integraler mellan imaginära gränsor. Svenska Vetenskaps–Akademiens Handlingar 6, Nr. 3, 1–18.

May, K.O. 1970. Émile Borel. DSB 1, 302–305.

Maz’ya, V. and T. Shaposhnikova. 1998. Jacques Hadamard, a universal mathematician. HMath 14. Providence, RI.

Mittag-Leffler, G. 1873. Försök tillett nytt bevis för en sats inom de definita integralernas teori. Öfversigt af K. Svenska Vetenskaps–Akademiens Förandlingar 30, 35–41.

Mittag-Leffler, G. 1875. Beweis für den Cauchy’schen Satz. Göttingen Nachr. 65–73.

Mittag-Leffler, G. 1884a. Sur la representation analytique des fonctions monogénes uniformes d’une variable indépendante. Acta 4, 1–79.MathSciNet

Mittag-Leffler, G. 1905. Sur une classe de functions entières. Verhandlungen des dritten internationalen Mathematiker–Kongresses, Heidelberg 1904, 258–264. Teubner, Leipzig.

Mittag-Leffler, G. 1909. Sur la répresentation arithmétique d’une fonction analytique d’une variable complexe. Atti del IV Congresso Internazionale dei Matematici, Roma 1908. Castelnuovo, G. (ed.). 1, 67–85. Tipografia Accademia dei Lincei, Roma.

Mittag-Leffler, G. 1923c. Der Satz von Cauchy über das Integral einer Funktion zwischen imaginären Grenzen. JfM 152, 1–5.MathSciNet

Montel, P. 1907. Sur les suites infinies de fonctions. Annales ENS (3) 4, 233–304.

Montel, P. 1910. Leçons sur les séries de polynomes à une variable complexe. Gauthier–Villars, Paris.MATH

Montel, P. 1912a. Sur les fonctions analytiques qui admettent deux valeurs exceptionnelles dans un domaine. CR 153, 996–998.

Montel, P. 1912b. Sur les familles de functions analytiques. Annales ENS (3) 29, 487–535.

Moore, E.H. 1900. A simple proof of the fundamental Cauchy–Goursat theorem. Trans. AMS 1, 499–506.MATHCrossRef

Mumford, D. 1984. Tata Lectures on Theta II. Birkhäuser, Boston.MATHCrossRef

Nabonnand, P. (ed.). 1999. La correspondence entre Henri Poincaré and Gösta Mittag-Leffler. Birkhäuser, Basel.MATH

Neumann, P.M. 2007. The concept of primitivity in group theory and the second memoir of Galois. AHES 60, 379–429.CrossRef

Nicholson, J. 1993. The development and understanding of the concept of quotient group. HM 20, 68–88.MathSciNetMATH

Noether, M. 1882. Note über die algebraischen Curven, welche eine Schaar eindeutiger Transformationen in sich zulassen. Math Ann. 20, 59–63, and Nachtrag zu dieser Note, Math Ann. 21, 138–140.

Osgood, W.F. 1900. On the existence of the Green’s function for the most general simply connected plane region. Trans. AMS 1, 310–314.MathSciNetMATH

Osgood, W.F. 1902b. Notes on the functions defined by an infinite series. Annals of mathematics (2) 3, 25–34.

Osgood, W.F. 1907. Lehrbuch der Funktionentheorie. vol. 1. Teubner, Leipzig. 2nd ed. Teubner, Leipzig 1912. 4th ed. Teubner, Leipzig 1923; vol. 2. Teubner, Leipzig 1924. 2nd ed. Teubner, Leipzig 1929–1932. Rep. Chelsea, New York 1965. Rep. AMS Chelsea, Providence, RI 1992.

Osgood, W.F. 1914. Topics in the theory of functions of several complex variables. The Madison colloquium 1913, II. AMS Colloquium Lectures, 4. Rep. Dover, New York 1966.

Osgood, W.F. and E.H. Taylor. 1913. Conformal transformations on the boundary of their regions of definition. Trans. AMS 14, 277–298.MathSciNetMATH

Parker, J. 2005. R.L. Moore: mathematician and teacher. Mathematical Association of America.

Parshall, K.H. and D.E. Rowe. 1994. The emergence of the American mathematical research community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore. HMath 8. Providence, RI.

Peano, G. 1890. Sur une courbe, qui remplit toute une aire plane. Math. Ann. 36, 157–160 in Op. scelte 1, 110–113.

Perron, O. 1952. Alfred Pringsheim. JDMV 56, 1–6.MathSciNetMATH

Picard, Ch.-E. 1883. Sur une proposition concernant les fonctions uniformes d’une variable lièes par une relation algébrique. Bull. sci. math. (2) 7, 107–116 in Oeuvres 1, 99–108.

Picard, E. 1890. Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J de math. (4) 6, 145–210 and 231 in Oeuvres 2, 385–450.

Picard, Ch.-E. 1912. Sur un théorème général relatif aux fonctions uniformes d’une variable liées par une relation algébrique. CR 154, 98–101 in Oeuvres 1, 217–221.

Pick, G. 1916. Über den Koebeschen Verzerrungssatz. Leipzig Berichte 68, 58–64.

Pincherle, S. 1897. Mémoire sur le calcul fonctionnel distributif. Math. Ann. 49, 325–382 in Opere scelte 2, 1–70.

Pincherle, S. 1906. Funktionaloperationen und–Gleichungen. EMW II A 11, 761–817. French trl. as Équations et opérations fonctionnelles. ESM II–26 (1912) 1–81. Not in Opere scelte.

Pincherle, S. 1918. Sull’iterazione della funzione x ² − 2. Rend. Lincei (5) 27, 337–343. Not in Opere scelte.

Pincherle, S. 1920a. L’iterazione completa di x ² − 2. Rend. Lincei (5) 29, 329–333. Not in Opere scelte.

Pincherle, S. 1920b. Sobre la iteración analítica. Revista matemática hispano–americana 2, 233–242; 265–271; 297–304. Not in Opere scelte.

Pincherle, S. 1925. Notice sur les travaux. Acta 46, 341–362 in Opere scelte 1, 45–63.

Plemelj, J. 1908b. Riemannsche Funktionenscharen mit gegebener Monodromiegruppe. Monatshefte Math. Phys. 19, 211–246.MathSciNetMATHCrossRef

Poincaré, H. 1881a. Sur les fonctions fuchsiennes. CR 92, 333–335 in Oeuvres 2, 1–4.

Poincaré, H. 1881b. Sur une nouvelle application et quelques propriétes importantes des fonctions fuchsiennes. CR 92, 859–861 in Oeuvres 1, 8–10.

Poincaré, H. 1881c. Sur les fonctions fuchsiennes. CR 92, 1274–1276 in Oeuvres 2, 16–18.

Poincaré, H. 1881d. Sur les fonctions fuchsiennes. CR 92, 1484–1487 in Oeuvres 2, 19–22.

Poincaré, H. 1882a. Sur les fonctions fuchsiennes. CR 94, 1038–1040 in Oeuvres 2, 41–43.

Poincaré, H. 1882b. Sur les transcendentes entières. CR 95, 23–26 in Oeuvres 4, 14–16.

Poincaré, H. 1882d. Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires. Math. Ann. 19, 553–564 in Oeuvres 1, 92–105.

Poincaré, H. 1883a. Sur les fonctions entières. Bull. SMF 11, 136–144 in Oeuvres 4, 17–24.

Poincaré, H. 1883d. Sur un théorème de la théorie générale des fonctions. Bull. SMF 11, 112–125 in Oeuvres 4, 57–69.

Poincaré, H. 1884a. Sur les groupes des équations linéaires. Acta 4, 201–312 in Oeuvres 2, 300–401.

Poincaré, H. 1885a. Sur les équations linéaires aux différentielles ordinaires et aux différences finies. Amer. J. Math. 7, 1–56 Oeuvres 1, 226–289.

Poincaré, H. 1885b. Sur un théorème de M. Fuchs, Acta 7, 1–32, in Oeuvres 3, 4–31.

Poincaré, H. 1898b. Les fonctions fuchsiennes et l’équation Δ u = e ^u. J de math. (5) 4, 137–230 in Oeuvres 2, 513–591.

Poincaré, H. 1907a. Sur l’uniformisation des fonctions analytiques. Acta 31, 1–63 in Oeuvres 4, 70–139.

Poincaré, H. 1908b. L’invention mathématique. In (Poincaré 1908a, 43–63).

Poincaré, H. 1997. Three supplementary essays on the discovery of Fuchsian functions. Gray J.J. and S.A. Walter (eds). Akademie Verlag, Berlin and Blanchard, Paris.

Porter, M.B. 1904. On functions defined by an infinite series of analytic functions of a complex variable. Annals of mathematics (2) 6, 45–48.

Pringsheim, A. 1893. Zur Theorie der Taylor’schen Reihe und der analytischen Functionen mit beschränktem Existenzbereich. Math. Ann. 42, 153–184.MathSciNetMATHCrossRef

Pringsheim, A. 1895a. Ueber den Cauchy’schen Integralsatz. Sitz. München 25, 39–72.

Pringsheim, A. 1898. Zur Theorie der Doppel–Integral. Sitz. München 28, 59–74.

Pringsheim, A. 1899a. Zur Theorie der Doppel–Integrale, Green’schen und Cauchy’schen Integralsatzes. Sitz. München 29, 39–62.

Pringsheim, A. 1899b. Grundlagen der allgemeinen Funktionenlehre. EMW II A 1, 1–53.

Pringsheim, A. 1912. Über einige funktionentheoretische Anwendungen der Eulerschen Reihen–Transformation. Sitz. München 42, 11–92.

Pringsheim, A. 1929. Kritisch–historische Bemerkungen zur Funktionentheorie, III. Sitzberichte München 59, 281–306.MATH

Radó, T. 1925. Über den Begriff der Riemannschen Fläche. Acta Szeged 2, 101–121.MATH

Remmert, R. 1998b. From Riemann surfaces to complex spaces. In Matériaux pour l’histoire des mathématiques au XX ^e siècle, 203–241. Societé Mathématique de France, Paris.

Richenhagen, G. 1985. Carl Runge (1856–1927): Von der reinen Mathematik zu Numerik. Vandenhoeck & Ruprecht, Göttingen.

Riemann, G.F.B. 1876b. Gleichgewicht der Electricität auf Cylindern mit kreisförmigem Querschnitt und parallel Axen. Conforme Abbildung von durch Kreise begrenzten Figuren. Ms. Werke, 472–476. Engl. trl. in (Riemann 2004, 429–434).

Runge, C. 1885a. Zur Theorie der eindeutigen analytischen Funktionen. Acta 6, 229–244.MathSciNetMATH

Sagan, H. 1994. Space–filling curves. Springer, New York.MATHCrossRef

Sagan, H. 2000. A geometrization of Lebesgue’s space–filling curve. In (Gray and Wilson 2000, 185–195).

Schafheitlin, P. 1908. Die Nullstellen der hypergeometrischen Funktion. Sitzungsberichte der Berliner Mathematische Gesellschaft 7, 19–28.

Schoenflies, A.M. 1896. Ueber einen Satz aus der Analysis situs. Göttingen Nachr. 79–89.

Scholz, E. 1999. The concept of a manifold, 1850–1950. In (James 1999, 25–64).

Schottky, F. 1877. Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen. JfM 83, 300–351.

Schottky, F. 1882. Ueber eindeutige Functionen mit linearen Transformationen in sich. (Auszug aus einem Schreiben an Herrn F. Klein). Math. Ann. 20, 299–300.

Schottky, F. 1906. Bemerkung zu meiner Mitteilung: Über den Picardschen Satz und die Borelschen Ungleichungen. Berlin Berichte, 32–36.

Schwarz, H.A. 1870c. Ueber die integration der partiellen Differentialgleichung \(\frac{{\partial }^{2}u} {\partial {x}^{2}} + \frac{{\partial }^{2}u} {\partial {y}^{2}} = 0\) unter vorgeschriebenen Grenz– und Unstetigkeitsbedingungen. Monatsberiche Berlin, 767–795 in Ges. Math. Abh. 2, 144–171.

Segal, S.L. 1981. Nine introductions to complex analysis. North Holland, Amsterdam.

Siegmund–Schultze, R. 1998. Eliakim Hasting Moore’s “general analysis”. AHES 52, 51–89.

Siegmund–Schultze, R. 2003. The origins of functional analysis. In (Jahnke 2003, 385–407).

Stäckel, P.G. 1893. Zur Theorie der eindeutigen analytischen Funktionen. JfM 112, 262–264.MATH

Study, E. 1913. Vorlesungen über ausgewählten Gegenstände der Geometrie. vol. 2. Konforme Abbildung einfach zusammenhängender Bereiche, unter Mitwirkung von W. Blaschke. Teubner, Leipzig.

Thomae, J. 1870a. Beitrag zur Bestimmung von \(\theta (0,0,\ldots 0)\) durch die Klassenmoduln algebraischer Funktionen. JfM 71, 201–222.

Valiron, G. 1949. Lectures on the general theory of integral functions. Chelsea, New York

Veblen, O. 1905. Theory of plane curves in non–metrical analysis situs. Trans. AMS 6, 83–98.MathSciNetMATHCrossRef

Vitali, G. 1903. Sopra le serie di funzioni analitiche. Rend. Lombardo (2) 36, 772–774 in Opere, 151–168.

Weierstrass, K.T.W. 1880c. Zur Funktionenlehre. Monatsberichte Berlin, 719–743. Nachtrag. Monatsberichte Berlin (1881) 228–230. Rep. in (Weierstrass 1886, 67–101, 102–104). In Math. Werke 2, 201–233. French trl. as: Remarques sur quelques points de la théorie des fonctions analytiques. Bull. sci. math. (2) 5 (1881) 157–183.

Weyl, H. 1913. Die Idee der Riemannschen Fläche. Teubner, Leipzig. 2nd ed. Teubner, Leipzig 1926. 3th ed. Teubner, Stuttgard 1955. New ed. Remmert, R. (ed.). Teubner, Leipzig 1997. Engl. trl. of 3th ed. as The concept of a Riemann surface. Addison–Wesley Pub. Co. Reading, Mass. 1955. Rep. Dover, New York 2009.

Weyl, H. 2008. Einführung in die Funktionentheorie. Meyer, R. and S. J. Patterson (eds). Birkhäuser, Basel.

Young, R.C. 1939. Schottky’s theorem. Gamble prize, 1939. Ms. University of Liverpool Archives D.599/12.

Zoretti, L. 1912. Les ensembles de points. In (Borel 1912, 113–170).

Titel: Chapter 8 Advanced Topics in the Theory of Functions
verfasst von: Umberto Bottazzini
Jeremy Gray
Verlag: Springer New York
Buch: Hidden Harmony—Geometric Fantasies
Print ISBN: 978-1-4614-5724-4

Electronic ISBN: 978-1-4614-5725-1

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-5725-1_9

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