References Books on Complex Analysis [Ah] Ahlfors, L. V.: Complex Analysis, 3rd edn. McCraw-Hill, New York 1979 [As] Ash, R. B.: Complex Variables, Academic Press, New York 1971 [BS] Behnke, H., Sommer, F.: Theorie der analytischen Funktionen einer kom- plexen Ver¨ anderlichen, 3. Aufl. Grundlehren der mathematischen Wis- senschaften, Bd. 77. Springer, Berlin Heidelberg New York 1965, Studi- enausgabe der 3. Aufl. 1976 [BG] Berenstein, C. A., Gay, R.: Complex Variables. An Introduction, Graduate Texts in Mathematics, vol. 125. Springer, New York Berlin Heidelberg 1991 [Bi] Bieberbach, L.: Lehrbuch der Funktionentheorie, Bd. I und II. Teubner, Leipzig 1930, 1931 — reprinted in Chelsea 1945, Johnson Reprint Corp. 1968 [Bu] Burckel, R. B.: An Introduction to Classical Complex Analysis, vol. I, Birk- h¨ auser, Basel Stuttgart 1979 (containing very detailed references) [Cara] Carath´ eodory, C.: Theory of Functions of a Complex Variable, (translated by F. Steinhardt) Vol. 1, Chelsea Publishing, New York, 1983. [CH] Cartan, H.: Elementary Theory of Analytic Functions of One or Several Complex Variables. Hermann, Paris and Addison-Wesley, Reading 1963 [Co1] Conway, J. B.: Functions of One Complex Variable, 2nd edn. 7th printing Graduate Texts in Mathematics, vol. 11. Springer, New York Heidelberg Berlin 1995 [Co2] Conway, J. B.: Functions of One Complex Variable II, corr. 2nd edn. Grad- uate Texts in Mathematics, vol. 159. Springer, New York Heidelberg Berlin 1995 [Din] Dinghas, A.: Vorlesungen ¨ uber Funktionentheorie, Grundlehren der mathe- matischen Wissenschaften, Bd. 110. Springer, Berlin Heidelberg New York 1961 [Ed] Edwards, H. M.: Riemann’s Zeta-Function, Academic Press, New York, London 1974
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References
Books on Complex Analysis
[Ah] Ahlfors, L.V.: Complex Analysis, 3rd edn. McCraw-Hill, New York 1979
[As] Ash, R.B.: Complex Variables, Academic Press, New York 1971
[BS] Behnke, H., Sommer, F.: Theorie der analytischen Funktionen einer kom-plexen Veranderlichen, 3. Aufl. Grundlehren der mathematischen Wis-senschaften, Bd. 77. Springer, Berlin Heidelberg New York 1965, Studi-enausgabe der 3. Aufl. 1976
[BG] Berenstein, C. A., Gay, R.: Complex Variables. An Introduction, GraduateTexts in Mathematics, vol. 125. Springer, New York Berlin Heidelberg 1991
[Bi] Bieberbach, L.: Lehrbuch der Funktionentheorie, Bd. I und II. Teubner,Leipzig 1930, 1931 — reprinted in Chelsea 1945, Johnson Reprint Corp.1968
[Bu] Burckel, R.B.: An Introduction to Classical Complex Analysis, vol. I, Birk-hauser, Basel Stuttgart 1979 (containing very detailed references)
[Cara] Caratheodory, C.: Theory of Functions of a Complex Variable, (translatedby F. Steinhardt) Vol. 1, Chelsea Publishing, New York, 1983.
[CH] Cartan, H.: Elementary Theory of Analytic Functions of One or SeveralComplex Variables. Hermann, Paris and Addison-Wesley, Reading 1963
[Co1] Conway, J. B.: Functions of One Complex Variable, 2nd edn. 7th printingGraduate Texts in Mathematics, vol. 11. Springer, New York HeidelbergBerlin 1995
[Co2] Conway, J. B.: Functions of One Complex Variable II, corr. 2nd edn. Grad-uate Texts in Mathematics, vol. 159. Springer, New York Heidelberg Berlin1995
[Din] Dinghas, A.: Vorlesungen uber Funktionentheorie, Grundlehren der mathe-matischen Wissenschaften, Bd. 110. Springer, Berlin Heidelberg New York1961
[Ed] Edwards, H.M.: Riemann’s Zeta-Function, Academic Press, New York,London 1974
[Gam] Gamelin, Theodore W.: Complex Analysis, 2nd corr. printing, Undergrad-uate Texts in Mathematics, Springer New York 2002
[Gre] Greene, R.E., Krantz, St.G.: Function Theory of One Complex Variable,2nd edition, AMS, Graduate Studies in Mathematics, vol. 40, Providence,Rhode Island 2002
[Hei] Heins, M.: Complex Function Theory, Academic Press, New York London1968
[HC] Hurwitz, A., Courant, R.: Funktionentheorie. Mit einem Anhang von H.Rohrl, 4.Aufl. Grundlehren der mathematischen Wissenschaften, Bd. 3.Springer, Berlin Heidelberg New York 1964
[How] Howie, J.H.: Complex Analysis, Springer, London 2003
[Iv] Ivic, A.: The Riemann Zeta-Function, Wiley, New York 1985
[Ja] Janich, K.: Funktionentheorie. Eine Einfuhrung, 6. Aufl. Springer-Lehrbuch, Springer, Berlin Heidelberg New York 2004
[LZL] Lu, J-K. L., Zhong, S-G., Liu, S-Q.: Introduction to the Theory of ComplexFunctions, Series in Pure Mathematics, vol 25, World Scientific, New Jersey,London, Singapore, Hong Kong, 2002
[Mar1] Markoushevich, A. I.: Theory of Functions of a Complex Variable, Prenti-ce-Hall, Englewood Cliffs 1965/1967
[MH] Marsden, J. E., Hoffmann, M. J.: Basic Complex Analysis, third edn., W.M.Freeman and Company, New York 1998
[McG] McGehee, O.Carruth: An Introduction to Complex Analysis, John Wiley& Sons, New York 2000
[Mo] Moskowitz, M.A.: A Course in Complex Analysis in One Variable, WorldScientific, New Jersey, London, Singapore, Hong Kong, 2002
[Na] Narasimhan, R.: Complex Analysis in One Variable, Birkhauser, BostonBasel Stuttgart 1985
[NP] Nevanlinna, R., Paatero, V.: Einfuhrung in die Funktionentheorie, Birk-hauser, Basel Stuttgart 1965
[Os1] Osgood, W.F.: Lehrbuch der Funktionentheorie I, II1, II2, Teubner, Leipzig1925, 1929, 1932
[Pal] Palka, B. P.: An Introduction to Complex Function Theory, UndergraduateTexts in Mathematics. 2nd corr. printing, Springer, New York 1995
References 511
[Pat] Patterson, S. T.: An Introduction to the Theory of Riemann’s Zeta–Function, Cambridge University Press, Cambridge 1988
[Re1] Remmert, R.: Theory of Complex Functions, Graduate Texts in Mathemat-ics, Readings in Mathematics, vol. 120, 1st. edn. 1991. Corr. 4th printing,Springer New York 1999
[ReS1] Remmert, R., Schumacher, G.: Funktionentheorie I, 5. Aufl. Springer-Lehrbuch, Springer, Berlin Heidelberg New York 2002
[ReS2] Remmert, R., Schumacher, G.: Funktionentheorie II, 3rd edn., Springer-Lehrbuch, Springer, Berlin Heidelberg New York 2005
[Ru] Rudin, W.: Real and Complex Analysis, 3rd edn. Mc Graw-Hill, New York1987
[Ap1] Apostol, T. M.: Modular Functions and Dirichlet Series in Number Theorie,2nd edn. Graduate Texts in Mathematics, vol. 41. Springer, New YorkBerlin Heidelberg 1992. Corr. 2nd printing 1997
[Ap2] Apostol, T.M.: Introduction to Analytic Number Theory, 2nd edn. Un-dergraduate Texts in Mathematics, Springer, New York Heidelberg Berlin1984. Corr. 5th printing 1998
[Ape] Apery, R.: Irrationalite de ζ(2) et ζ(3), Asterisque 61, pp. 11-13, 1979
[Ch1] Chandrasekharan, K.: Introduction to Analytic Number Theory,Grundlehren der mathematischen Wissenschaften, Bd. 148. Springer,Berlin Heidelberg New York 1968
[Ch2] Chandrasekharan, K.: Elliptic Functions, Grundlehren der mathematischenWissenschaften, Bd. 281. Springer, Berlin Heidelberg New York 1985
[Ch3] Chandrasekharan, K.: Arithmetical Functions, Grundlehren der mathema-tischen Wissenschaften, Bd. 167. Springer, Berlin Heidelberg New York1970
[CS] Conway, J. H., Sloane, N. J. A.: Sphere Packings, Lattices and Groups. 2ndedn. Grundlehren der mathematischen Wissenschaften 290. Springer, NewYork Berlin Heidelberg 1999
[DS] Diamond, F., Shurman, J.: A First Course in Modular Forms, GraduateTexts in Mathematics, vol. 228, Springer 2005
[Fr1] Fricke, R.: Die elliptischen Funktionen und ihre Anwendungen, first part:Teubner, Leipzig 1916, second part: Teubner, Leipzig 1922. Reprinted byJohnson Reprint Corporation, New York London 1972
[Fo] Forster, O.: Lectures on Riemann Surfaces. Graduate Texts in Mathemat-ics, vol. 81, Springer, Berlin Heidelberg New York 1981 (2nd corr. printing1991)
[Ga] Gaier, D.: Konstruktive Methoden der konformen Abbildung . SpringerTracts in Natural Philosophy, vol. 3. Springer, Berlin Heidelberg New York1964
[Gu] Gunning, R.C.: Lectures on Modular Forms. Annals of Mathematics Stud-ies, No 48. Princeton University Press, Princeton, N. J., 1962
[He1] Hecke, E.: Lectures on Dirichlet Series, Modular Functions and QuadraticForms, Vandenhoeck & Ruprecht, Gottingen 1983
[Hen] Henrici, P.: Applied and computational complex analysis, vol. I, II, III .Wiley, New York 1974, 1977, 1986
[Iw] Iwaniec, H.: Topics in Classical Automorphic Forms. AMS, Graduate Stud-ies in Mathematics, vol. 17, 1997
[JS] Jones, G. A., Singerman, D.: Complex Functions, an Algebraic and Geo-metric Viewpoint . Cambridge University Press, Cambridge 1987
[KK] Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen, 2. Aufl.Springer, Berlin Heidelberg 2007
[Ko] Koblitz, N.: Introduction to Elliptic Curves and Modular Forms, 2nd edn.Graduate Texts in Mathematics, vol. 97. Springer, New York Berlin Hei-delberg 1993
[Lan] Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen, Bd. I,Bd. II . Teubner, Leipzig 1909; 3rd edn. Chelsea Publishing Company, NewYork 1974
[La2] Lang, S.: Algebra, 4rd corr. printing, Graduate Texts in Mathematics 211,Springer New York 2004
[Le] Leutbecher, A.: Vorlesungen zur Funktionentheorie I und II, Mathematis-ches Institut der Technischen Universitat Munchen (TUM) 1990, 1991
[Ma2] Maaß, H.: Funktionentheorie II, III, Vorlesungsskript, Mathematisches In-stitut der Universitat Heidelberg 1949
[Ma3] Maaß, H.: Modular Functions of one Complex Variable, Tata Institute ofFundamental Research, Bombay 1964. Revised edn.: Springer, Berlin Hei-delberg New York 1983
[Mi] Miyake, T.: Modular forms, Springer, Berlin Heidelberg New York 1989
[Mu] Mumford, D.: Tata Lectures on Theta I, Progress in Mathematics, vol. 28.Birkhauser, Boston Basel Stuttgart 1983
[Ne1] Nevanlinna, R.: Uniformisierung, 2. Aufl. Grundlehren der mathematischenWissenschaften, Bd. 64. Springer, Berlin Heidelberg New York 1967
[Ne2] Nevanlinna, R.: Eindeutige analytische Funktionen, 2. Aufl. Grundlehrender mathematischen Wissenschaften, Bd. 46. Springer, Berlin HeidelbergNew York 1974 (reprint)
References 513
[Pf] Pfluger, A.: Theorie der Riemannschen Flachen, Grundlehren der mathe-matischen Wissenschaften, Bd. 89. Springer, Berlin Gottingen Heidelberg1957
[Po] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps, SpringerBerlin 1992
[Pr] Prachar, K.: Primzahlverteilung, 2. Aufl. Grundlehren der mathematischenWissenschaften, Bd. 91. Springer, Berlin Heidelberg New York 1978
[Ra] Rankin, R.A.: Modular Forms and Functions. Cambridge University Press,Cambridge, Mass., 1977
[Ro] Robert, A.: Elliptic Curves. Lecture Notes in Mathematics, vol. 326 (2ndcorr. printing). Springer, Berlin Heidelberg New York, 1986
[Sb] Schoeneberg, B.: Elliptic Modular Functions. Grundlehren der mathemati-schen Wissenschaften, Bd. 203. Springer, Berlin Heidelberg New York 1974
[Sch] Schwarz, W.: Einfuhrung in die Methoden und Ergebnisse der Primzahltheo-rie, B I-Hochschultaschenbucher, Bd. 278/278a. Bibliographisches Institut,Mannheim Wien Zurich 1969
[Se] Serre, J. P.: A Course in Arithmetic, Graduate Texts in Mathematics, vol.7. Springer, New York Heidelberg Berlin 1973 (4th printing 1993)
[Sh] Shimura, G.: Introduction to Arithmetic Theory of Automorphic Functions,Publications of the Mathematical Society of Japan 11. Iwanami Shoten,Publishers and Princeton University Press 1971
[Si1] Siegel, C. L.: Topics in Complex Function Theory, vol. I, II, III . Intersc.Tracts in Pure and Applied Math., No 25. Wiley-Interscience, New York1969, 1971, 1973
[Sil] Silverman, J. H.: Advanced Topics in the Arithmetic of Elliptic Curves,Graduate Texts in Mathematics, vol. 151 Springer, New York Berlin Hei-delberg 1994
[ST] Silverman, J., Tate, J.: Rational Points on Elliptic Curves, UndergraduateTexts in Mathematics, Springer, New York Berlin Heidelberg 1992
[Sp] Springer, G.: Introduction to Riemann Surfaces, Addison-Wesley, Reading,Massachusetts, USA 1957
[Tit2] Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, ClarendonPress, Oxford 1951, reprinted 1967
[We] Weil, A.: Elliptic Functions according to Eisenstein and Kronecker, Ergeb-nisse der Mathematik und ihrer Grenzgebiete, Bd. 88. Springer, Berlin Hei-delberg New York 1976
[WK] Weierstraß, K.: Einleitung in die Theorie der analytischen Funktionen, Vor-lesung, Berlin 1878. Vieweg, Braunschweig Wiesbaden 1988
[WH] Weyl, H.: Die Idee der Riemannschen Flache, 4. Aufl. Teubner, Stuttgart1964, new edition 1997, editor R. Remmert
History of the Complex Numbers and ComplexFunctions
[Bel] Belhoste, B.: Augustin-Louis Cauchy. A Biography, Springer, New YorkBerlin Heidelberg 1991
514 References
[CE] Cartan, E.: Nombres complexes, Expose, d’apres l’article allemand de E.Study (Bonn). Encyclop. Sci. Math. edition francaise I 5, p. 329–468.Gauthier-Villars, Paris; Teubner, Leipzig 1909; see also E.Cartan: Œvrescompletes II.1, p. 107–246, Gauthier-Villars, Paris 1953
[Die2] Dieudonne, J. (Ed.): Abrege d’histoire des mathematiques I & II, HermannParis 1978
[Eb] Ebbinghaus, H.-D. et al.: Numbers, 3rd. corr. printing, Graduate Texts inMathematics 123, Springer New York 1996 Springer-Lehrbuch, Springer,Berlin Heidelberg New York 1992
[Fr2] Fricke, R.: IIB3. Elliptische Funktionen. Encyklopadie der mathematischenWissenschaften mit Einschluß ihrer Anwendungen, Bd. II 2, Heft 2/3, S.177–348. Teubner, Leipzig 1913
[Fr3] Fricke, R.: IIB4. Automorphe Funktionen mit Einschluß der elliptischenFunktionen. Encyklopadie der mathematischen Wissenschaften mit Ein-schluß ihrer Anwendungen, Bd. II 2, Heft2/3, S. 349–470. Teubner, Leipzig1913
[Hi] Hirzebruch, F.: chapter 11 in [Eb]
[Hou] Houzel, C.: Fonctions elliptiques et integrales abeliennes, chap. VII, pp. 1–113 in [Die2], vol. II
[Kl] Klein, F.: Vorlesungen uber die Entwicklung der Mathematik im 19. Jahr-hundert, Teil 1 und 2, Grundlehren der mathematischen Wissenschaften,Bd. 24 und 25. Springer, Berlin Heidelberg 1926. Nachdruck in einem Band1979
[Mar2] Markouschevitsch, A. I.: Skizzen zur Geschichte der analytischen Funktio-nen, Hochschultaschenbucher fur Mathematik, Bd. 16. Deutscher Verlagder Wissenschaften, Berlin 1955
[Neu] Neuenschwander, E.: Uber die Wechselwirkung zwischen der franzosischenSchule, Riemann und Weierstraß. Eine Ubersicht mit zwei Quellenstudien.Arch. Hist. Exact Sciences 24 (1981), 221–255
[Os2] Osgood, W.F.: Allgemeine Theorie der analytischen Funktionen a) einerund b) mehrerer komplexer Großen, Enzyklopadie der MathematischenWissenschaften, Bd. II 2, S. 1–114. Teubner, Leipzig 1901–1921
[Re2] Remmert, R.: Complex Numbers, Chap. 3 in [Eb]
[St] Study, E.: Theorie der gemeinen und hoheren complexen Grossen, Enzyklo-padie der Mathematischen Wissenschaften, Bd. I 1, S. 147–183. Teubner,Leipzig 1898–1904
[Ver] Verley, J. L.: Les fonctions analytiques, Chap IV, pp. 129–163 in [Die2], vol.I
In [ReS1] and [ReS2] one can find many facts related to the history of thetheory of complex functions.
Original Papers
[Ab1] Abel, N.H.: Memoire sur une propriete generale d’une classe tres etenduede fonctions transcendantes (submitted at 30. 10. 1826, published in 1841).
References 515
Œvres completes de Niels Henrik Abel, tome premier, XII, p. 145–211.Grondahl, Christiania M DCCC LXXXI, Johnson Reprint Corporation1973
[Ab2] Abel, N.H.: Recherches sur les fonctions elliptiques, Journal fur die reineund angewandte Mathematik 2 (1827), 101–181 und 3 (1828), 160–190; seealso Œvres completes de Niels Henrik Abel, tome premier, XVI, p. 263–388. Grondahl, Christiania M DCCC LXXXI, Johnson Reprint Corporation1973
[Ab3] Abel, N.H.: Precis d’une theorie des fonctions elliptiques, Journal fur diereine und angewandte Mathematik 4 (1829), 236–277 und 309–370; seealso Œvres completes de Niels Henrik Abel, tome premier, XXVIII, p. 518–617. Grondahl, Christiania M DCCC LXXXI, Johnson Reprint Corporation1973
[BFK] Busam, R., Freitag, E., Karcher, W.: Ein Ring elliptischer Modulformen,Arch. Math. 59 (1992), 157–164
[Cau] Cauchy, A.-L.: Abhandlungen uber bestimmte Integrale zwischen imaginarenGrenzen. Ostwald’s Klassiker der exakten Wissenschaften Nr. 112, WilhelmEngelmann, Leipzig 1900; see also A.-L.Cauchy: Œuvres completes 15, 2.Ser., p. 41–89, Gauthier-Villars, Paris 1882–1974The source appeared as “Memoire sur les integrales definies, prises entredes limites imaginaires” in 1825.
[Dix] Dixon, J.D.: A brief proof of Cauchy’s integral theorem, Proc. Am. Math.Soc. 29 (1971), 635–636
[Eis] Eisenstein, G.: Genaue Untersuchung der unendlichen Doppelproducte, auswelchen die elliptischen Functionen als Quotienten zusammengesetzt sind,und der mit ihnen zusammenhangenden Doppelreihen (als eine neue Be-grundungsweise der Theorie der elliptischen Functionen, mit besondererBerucksichtigung ihrer Analogie zu den Kreisfunctionen). Journal fur diereine und angewandte Mathematik (Crelle’s Journal) 35 (1847), 153–274;see also G.Eisenstein: Mathematische Werke, Bd. I. Chelsea PublishingCompany, New York, N. Y., 1975, S. 357–478
[El] Elstrodt, J.: Eine Charakterisierung der Eisenstein-Reihe zur SiegelschenModulgruppe, Math. Ann. 268 (1984), 473-474
[He2] Hecke, E.: Uber die Bestimmung Dirichletscher Reihen durch ihre Funktion-algleichung, Math. Ann. 112 (1936), 664–699; see also E. Hecke: Mathema-tische Werke, 3. Aufl., S. 591–626. Vandenhoeck & Ruprecht, Gottingen1983
[He3] Hecke, E.: Die Primzahlen in der Theorie der elliptischen Modulfunktionen,Kgl. Danske Videnskabernes Selskab. Mathematisk-fysiske Medelelser XIII,10, 1935; see also E.Hecke: Mathematische Werke, S. 577–590. Vandenhoeck& Ruprecht, Gottingen 1983
[Hu1] Hurwitz, A.: Grundlagen einer independenten Theorie der elliptischen Mo-dulfunktionen und Theorie der Multiplikator-Gleichungen erster Stufe, In-auguraldissertation, Leipzig 1881; Math. Ann. 18 (1881), 528–592; seealso A.Hurwitz: Mathematische Werke, Band I Funktionentheorie, S. 1–66, Birkhauser, Basel Stuttgart 1962
516 References
[Hu2] Hurwitz, A.: Uber die Theorie der elliptischen Modulfunktionen, Math.Ann. 58 (1904), 343–460; see also A.Hurwitz: Mathematische Werke, BandI Funktionentheorie, S. 577–595, Birkhauser, Basel Stuttgart 1962
[Ig1] Igusa, J.: On the graded ring of theta constants, Amer. J. Math. 86 (1964),219–246
[Ig2] Igusa, J.: On the graded ring of theta constants II, Amer. J. Math. 88(1966), 221–236
[Ja1] Jacobi, C.G. J.: Suite des notices sur les fonctions elliptiques, Journal furdie reine und angewandte Mathematik 3 (1828), 303–310 und 403–404; seealso C. G. J. Jacobi’s Gesammelte Werke, I, S. 255–265, G.Reimer, Berlin1881
[Ja2] Jacobi, C. G. J.: Fundamenta Nova Theoriae Functionum Ellipticarum,Sumptibus fratrum Borntrager, Regiomonti 1829; see also C. G. J. Jacobi’sGesammelte Werke, I, S. 49–239, G.Reimer, Berlin 1881
[Ja3] Jacobi, C.G. J.: Note sur la decomposition d’un nombre donne en quatrequarres, C.G. J. Jacobi’s Gesammelte Werke, I, S. 274, G. Reimer, Berlin1881
[Ja4] Jacobi, C. G. J.: Theorie der elliptischen Funktionen, aus den Eigenschaftender Thetareihen abgeleitet, after a lecture of Jacobi, revised at his request byC.Borchardt. C.G. J. Jacobi’s Gesammelte Werke, I, S. 497–538, G.Reimer,Berlin 1881
[Re3] Remmert, R.: Wielandt’s Characterization of the Γ -function, pp. 265–268in [Wi]
[Ri1] Riemann, B.: Grundlagen fur eine allgemeine Theorie der Functionen einerveranderlichen complexen Grosse, Inauguraldissertation, Gottingen 1851;see also B. Riemann: Gesammelte mathematische Werke, wissenschaftlicherNachlaß und Nachtrage, collected papers, S. 35–77. Springer, Berlin Hei-delberg New York; Teubner, Leipzig 1990
[Ri2] Riemann, B.: Ueber die Anzahl der Primzahlen unterhalb einer gegebe-nen Grosse, Monatsberichte der Berliner Akademie, November 1859, S.671–680; see also B.Riemann: Gesammelte mathematische Werke, wis-senschaftlicher Nachlaß und Nachtrage, collected papers, S. 177–185.Springer, Berlin Heidelberg New York, Teubner, Leipzig 1990
[Si2] Siegel, C. L.: Uber die analytische Theorie der quadratischen Formen, Ann.Math. 36 (1935), 527–606; see also C. L. Siegel: Gesammelte Abhandlungen,Band I, S. 326–405. Springer, Berlin Heidelberg New York 1966
[Wi] Wielandt, H.: Mathematische Werke, vol 2, de Gruyter, Berlin New York1996
Collections of Exercises
Parallely to the problem books among the Knopp [Kno] editions we espe-cially recommend:
[Kr] Krzyz, J.G.: Problems in Complex Variable Theory, Elsevier, New YorkLondon Amsterdam 1971
[Sha] Shakarchi, R.: Problems and Solutions for Complex Analysis. Springer, NewYork Berlin Heidelberg 1995(In this book one can find solutions to all exercises in Lang’s book [La1].)
[Tim] Timmann, S.: Repetitorium der Funktionentheorie, Verlag Binomi, Springe1998
and also the classical
[PS] Polya, G. Szego, G.: Problems and Theorems in Analysis II, Theory ofFunctions, Zeros, Polynomials, Determinants, Number Theory, Geometry,Classics in Mathematics, Springer 1998, Reprint of the 1st ed. Berlin, Hei-delberg, New York 1976
Symbolic Notations
iff if and only ifL.H.S. left hand sideR.H.S. right hand sideN = { 1, 2, . . . } set of natural numbersN0 = { 0, 1, 2, . . . } set of natural numbers including zeroZ ring of integersR field of real numbers, real axisC field of complex numbers, complex planeC− = C \ { x ∈ R ; x ≤ 0 } slit plane along the negative
real half-lineC
• = C \ {0} punctured plane
C = C ∪ {∞} Riemann spherePn(C) n-dimensional projective spaceH upper half-planeE open unit diskS1 unit circleH Hamiltonian quaternionsRe z , Im z real and imaginary part of a number zRe f , Im f real and imaginary part of a function fz complex conjugate of z|z| modulus, absolute value of zArg z (−π < Arg z ≤ π) principal value of the argumentLog z = log |z|+ iArg z principal value of the logarithm◦D set of interior points in D
A closure of AJ(f, a) : C→ C Jacobi map of f in aΔ = ∂2
1 + ∂22 Laplace operator∫
αf Line integral of f along the curve α
l(α) length of the piecewise smooth curve α
520 Symbolic Notations
α⊕ β composition of two curvesα− inverse (reciprocal) curve〈z1, z2, z3〉 triangular path
Ur(a) , Ur(a) open resp. closed disk centered at a with radius r∮f integral along a circleO(D) ring of analytic functions on DA annular domainA(a; r,R) annular domain with center a and radii r,Rχ(α; a) index of the closed curve α around aRes(f ; a) residue of f in aInt(α) interior of the closed curve αExt(α) exterior of the closed curve αS2 unit sphere in R
3
M group of Mobius transformationsAut(D) group of conformal self-maps of DM(D) field of meromorphic functions on a domain DCR(z, a, b, c) cross ratioΓ (z) , Γ (s) gamma functionB(z,w) beta function℘ ℘-function of Weierstrass
Gk Eisenstein series of weight kg2 , g3 g2 = 60G4 , g3 = 140G6 ,K(L) field of elliptic function for the lattice LK(Γ ) field of elliptic modular functions for
the modular group Γσ(z) Weierstrass’ σ-functionϑ(τ, z) , ϑ(z,w) Jacobi theta functionj(τ ) absolute invariantΔ(τ ) discriminantSL(2,R) group of real 2× 2 matrices with determinant 1Γ = SL(2,Z) elliptic modular group[Γ, k] vector space of all modular forms of weight k[Γ, k]0 ⊂ [Γ, k] vector space of all cusp forms of weight kF fundamental region of the modular groupΓϑ theta groupFϑ fundamental region of the theta groupΓ [q] principal congruence group of level qΘ(x) , ψ(x) Tschebyscheff functionsπ(x) prime number functionLi(x) integral logarithmζ(s) Riemann zeta function
sector 144Apery, R. (1916-1994) 186arc length 72arcwise connected 77argument 15Argument Principle 172associative law 10automorphism
conformal 225of a domain 232of H 303, 308
automorphism groupof H 308of the complex plane 160of the Riemann sphere 160
automorphy factor 355
Babylonian identity 346bank 435Bernoulli, Jacob (1654-1705) 185Bernoulli number 185Bessel, F.W. (1784-1846) 69, 154Bessel differential equation 154Big Theorem of Picard 139bilinear form
standard ∼ in Cn 348
522 Index
Binet, J.P.M (1786-1856) 154binomial formula 16binomial series 33Bohr, H. (1887-1951) 209Bolzano, B.(1781-1848) 40Bombelli, R. (1526-1573) 1Borel, E. (1871-1956) 39, 105branch of the logarithm 86bridge 408
calculation with complex powers 30Caratheodory, C. (1873-1950) 130Cardano, G. (1501-1576) 1Casorati, F. (1835-1890) 333Cauchy, A.L. (1789-1857) 19, 26,
69, 84, 111, 114, 115, 118, 165, 233Cauchy
estimate 123estimates 147sequence 31
Cauchy-Hadamard formula 122Cauchy Integral Formula
for annuli 144generalized 244generalized 96
Cauchy Integral Theorem 81for star domains 84for triangular paths 81generalized 244homological version 240homotopical version 233, 236
Cauchy principal value 177Cauchy-Riemann equations 47, 50,
64Cauchy’s multiplication theorem for
series 26Cauchy-Schwarz inequality 19Cayley, A. (1821-1895) 23, 67, 233Cayley
map 67, 233numbers 23
chain rule 44character 66, 355, 374
principal 355character relation 374characterization of ϑr 389Chinese Remainder Theorem 368classes
of cusps 354
classification of singularities 138Clausen, Th. (1801-1885) 36clockwise 74closed 39, 40closure 40coincidence set 125, 126commensurable 267commutative law 10commutator group 372compact 39
sequence 40compactly convergent 105complex
derivative 42differentiable 42integral 69line integral 69
permanence properties 44complex exponential function 26complex number field 21, 23complex numbers 1, 9, 10, 22complex sine 26computation of improper integrals 177computation of integrals using the
elliptic modular form 317elliptic modular group 301, 304, 308,
325generators 308
elliptic modular group 317, 318Elstrodt, J. 392, 407entire
modular form 331entire function 97entire modular form 331Epstein, P. (1871-1939) 447Epstein
ζ-function 447equation
differential 55Differential ∼ of ℘ 251
equationsCauchy-Riemann 47
equivalenceof unimodular matrices 346
equivalence classof cusps 353of lattices 301, 305
equivalenttopologically 224, 237
equivalent lattices 301Erdos, P. (1913-1996) 446error estimates 425Euclid (≈ 300 before our era) 444Euler, L. (1707-1783) 1, 186, 192,
199, 200, 206, 288, 403, 404, 422,434, 444
Euler
Addition Theorem 251beta function 208indicator function ϕ 428Pentagonal Number Theorem 399product 404, 422product of the ζ-function 404∼’s Product Formula 207, 404
Euler-Mascheroni’s Constant 200,206, 434
Euler numbers 123Eulerian integral of the second kind
192exchange
of differential with limit 106of differentiation and summation
106of integral with limit 105
Existence theoremfor analytic logarithms 86for analytic roots 86
exp 26, 27kernel of ∼ 28
exponential function 26–28, 54, 66characterization 66
exponentiation laws 31exterior part of a closed curve 163
factor group 251factor ring 23factorization 403Fagnano, G.C. (1682-1766) 251, 288Fermat, P. de (1601-1665) 18Fermat prime 18Fibonacci, L.P. (1170
even 268entire 97, 158η of Dedekind 398exponential 26–28Γ 221Γ 192harmonic 55, 244inverse 38, 251, 284j 306, 307, 311, 313, 397Fourier series 398
Joukowski 66Mangoldt 423meromorphic 155, 158, 252modular 252μ of Mobius 427, 446℘ 266, 268℘ of Weierstrass 217, 298periodic
period 1 152ϕ of Euler 428potential 55ψ 423ψ of Gauss 208ψ of Tschebyscheff 424rational 155, 158σ 294σ of Weierstrass 216, 300sine 26τ of Ramanujan 398, 407, 419theta 251Θ of Tschebyscheff 424Tschebyscheff 423Weierstrass ℘ ∼
half lattice values 265Weierstrass ℘ 221, 268ζ 417ζ of Riemann 400, 404ζ of Weierstrass 216, 298
functional determinant 50functional equation
for the Dirichlet series 409for the Epstein ζ-function 447for the exponential function 27for the Γ -function 195
526 Index
for the ζ-function 417, 434, 435fundamental domain 318, 333
for the theta group 364, 365of the theta group 390volume 369
fundamental parallelogram 254fundamental region 254
for the modular group 312Fundamental Theorem of Algebra 9,
18, 90, 97, 129, 133, 174
γ (constant) 434Γ -function 192, 221
Prym’s decomposition 221characterization 195, 207completion formula 201duplication formula 202functional equation 195growth 205, 412product formula 196product representation 200
gaussian number plane 13generalized circle 161generators of the theta group 356geometric series 25globally conformal 87Goursat E.J.-B. (1858-1937) 81group 302
absolutely convergent 197for the sine 201, 215normally convergent 198of the ζ-function 404
infinite series 25infinity place 252integrable 70integral 69
contour 72elliptic ∼ of first kind 284Fresnel 91gamma 193improper 177line 69, 72Mellin 412path 234standard estimate 73
integral formula 93integral logarithm 426integral representation
Hankel’s ∼ for 1/Γ 209integral ring 127integration
partial 71rule 69
integrity domain 127, 217interior part of a closed curve 163interior point 40interval 77inverse function 38inversion at the unit circle 22irreducible 23
polynomial ring 337, 378potential functions 55power series 109
rearrangement 117prime 18
element in a ring 217prime number 402, 444prime number distribution 421prime number theorem 424, 444primitive 69, 79, 84, 244primitive nth root of unity 22primitive root of unity 22principal branch of the logarithm 29,
55, 75, 85principal character 355principal congruence subgroup 352
of level two 366generators 367
principal part 143principal part distribution 220principal value of the argument 15, 30,
38principal value of the logarithm 38,
197principle
Argument P∼ 172Pringsheim A. (1850-1941) 81problem
extremal value 228product formula
Wallis 217projective closure of a curve 276projective space 273, 274
finite part 274infinite part 274
Prym, E.F. (1841-1915) 221punctured disk 133purely imaginary 12
quadratic form 342representation number 381
quotient field 156
530 Index
quotient topology 334
Ramanujan, S.A. (1887-1920) 349,407
Ramanujan
Conjecture 398τ -function 398, 407, 419
ramification point 258for ℘ 265
rational function 100, 155real analytic 127real part 12rearrangement 385rectangular lattice 278reflection at the unit circle 22regular 53, 326, 416representation numbers for quadratic
forms 381residue 165, 170
computation rules 167formula 166transformation formula 170