Selected Topics in Complex Analysis

In this chapter we will discuss several topics in Complex Analysis which usually are not or only incomprehensively considered in lectures and textbooks, but are of particular interest in the field of Analytic and Algebraic Differential Equations. Our standard reference is Ahlfors’ forever young Complex Analysis.

[1]  A. Z. Mokhon’ko,et al.  Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations , 1974 .

[2]  A. Ros.  THE GAUSS MAP OF MINIMAL SURFACES , 2002 .

[3]  F. Brüggemann.  PROOF OF A CONJECTURE OF FRANK AND LANGLEY CONCERNING ZEROS OF MEROMORPHIC FUNCTIONS AND LINEAR DIFFERENTIAL POLYNOMIALS , 1992 .

[4]  Defect relation for rational functions as targets , 2005 .

[5]  Walter Bergweiler,et al.  On the singularities of the inverse to a meromorphic function of finite order , 1995 .

[6]  A. Eremenko,et al.  Nevanlinna functions with real zeros , 2003, math/0310496.

[7]  Iterations of rational functions and the distribution of the values of the Poincaré functions , 1992 .

[8]  Nalini Joshi,et al.  On Boutroux's Tritronquée Solutions of the First Painlevé Equation , 2001 .

[9]  E. Hille.  Some remarks on Briot-Bouquet differential equations, II , 1978 .

[10]  J. B. McLeod,et al.  Integral Equations and Connection Formulae for the Painlevé Equations , 1992 .

[11]  N. Steinmetz.  Value distribution of the Painlevé Transcendents , 2002 .

[12]  LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONAL FUNDAMENTAL SETS , 1991 .

[13]  N. Steinmetz.  Remark on meromorphic functions sharing five pairs , 2015 .

[14]  E. Mues.  Zur Faktorisierung elliptischer Funktionen , 1971 .

[15]  J. M. Whittaker.  The Order of the Derivative of a Meromorphic Function , 1936 .

[16]  A. Eremenko,et al.  Meromorphic solutions of higher order Briot–Bouquet differential equations , 2007, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  S. Shimomura.  LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS , 2004, Proceedings of the Edinburgh Mathematical Society.

[18]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[19]  G. Weiss,et al.  EIGENFUNCTION EXPANSIONS. Associated with Second-order Differential Equations. Part I. , 1962 .

[20]  R. Kaufman,et al.  On the order of growth of meromorphic solutions of first-order differential equations , 1979 .

[21]  P. Boutroux,et al.  Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre (suite) , 1913 .

[22]  Henrik L. Selberg.  Über die Wertverteilung der algebroiden Funktionen , 1930 .

[23]  S. P. Hastings,et al.  A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation , 1980 .

[24]  Linear differential equations with exceptional fundamental sets. II , 1993 .

[25]  Norbert Steinmetz.  Zur Theorie der binomischen Differentialgleichungen , 1979 .

[26]  On the distribution of zeros of solutions of second-order differential equations , 1989 .

[27]  J. Malmquist,et al.  Sur les fonctions à un nombre fini de branches satisfaisant à une équation différentielle du premier ordre , 1920 .

[28]  Ye Yasheng,et al.  On the Zeros of a Differential Polynomial and Normal Families , 1997 .

[29]  T. Kecker.  A Cubic Hamiltonian System with Meromorphic Solutions , 2016 .

[30]  N. Steinmetz.  COMPLEX RICCATI DIFFERENTIAL EQUATIONS REVISITED , 2014 .

[31]  G. Frank.  Picardsche Ausnahmewerte bei Lösungen linearer Differentialgleichungen , 1970 .

[32]  G. G. Gundersen.  Meromorphic functions that share five pairs of values , 2011 .

[33]  S. Shimomura.  Proofs of the Painlevé property for all Painlevé equations , 2003 .

[34]  S. Shimomura.  Growth of the first, the second and the fourth Painlevé transcendents , 2003, Mathematical Proceedings of the Cambridge Philosophical Society.

[35]  H. Wittich.  Neuere Untersuchungen über eindeutige analytische Funktionen , 1955 .

[36]  X. Pang.  ON NORMAL CRITERION OF MEROMORPHIC FUNCTIONS , 1990 .

[37]  Kazuo Okamoto.  On the τ-function of the Painlevé equations , 1981 .

[38]  Norbert Steinmetz.  Über eine Klasse von Painlevéschen Differentialgleichungen , 1983 .

[39]  A. Hinkkanen,et al.  Solutions of a modified third Painlevé equation are meromorphic , 2001 .

[40]  G. G. Gundersen.  Meromorphic functions that share three values IM and a fourth value CM , 1992 .

[41]  R. Nevanlinna,et al.  Le theoreme de Picard-Borel et la theorie des fonctions meromorphes , 1930, The Mathematical Gazette.

[42]  Fritz Gackstatter,et al.  Zur Theorie der gewohnlichen Differentialgleichungen im Komplexen , 1980 .

[43]  S. Krantz.  Function theory of several complex variables , 1982 .

[44]  A. Hinkkanen,et al.  The meromorphic nature of the sixth Painlevé transcendents , 2004 .

[45]  J. Langley.  On the zeros of the second derivative , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[46]  Yurii Lyubarskii,et al.  Lectures on entire functions , 1996 .

[47]  N. Steinmetz.  Exceptional values of solutions of linear differential equations , 1989 .

[48]  W. Hayman,et al.  The Strength of Cartan's Version of Nevanlinna Theory , 2004 .

[49]  G. Frank,et al.  Über die Nullstellen von Lösungen linearer Differentialgleichungen , 1983 .

[50]  C. Osgood.  Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better , 1985 .

[51]  T. Shimizu.  On the theory of meromorphic functions , 1929 .

[52]  W. Hayman.  The Local Growth of Power Series: A Survey of the Wiman-Valiron Method , 1974, Canadian Mathematical Bulletin.

[53]  Olli Lehto,et al.  Boundary behaviour and normal meromorphic functions , 1957 .

[54]  A. Its,et al.  Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution , 1998 .

[55]  N. Steinmetz.  Sub‐normal solutions to Painlevé's second differential equation , 2011, 1108.2188.

[56]  G. Pólya.  Über den Zusammenhang Zwischen dem Maximalbetrage Einer Analytischen Funktion und Dem Grössten Gliede der Zugehörigen Taylorschen Reihe , 1916 .

[57]  L. Sons.  Deficiencies of monomials , 1969 .

[58]  N. Steinmetz.  ON THE ZEROS OF (f(p) + ap-1f(p-1)+ ... + a0f)f , 1987 .

[59]  K. Yosida.  On algebroid-solutions of ordinary differential equations , 1933 .

[60]  J. Langley.  An inequality of Frank, Steinmetz and Weissenborn , 2011 .

[61]  P. Hartman.  Ordinary Differential Equations , 1965 .

[62]  G. Weissenborn.  On the Theorem of Tumura and Clunie , 1986 .

[63]  Über eine Defekt- und Verzweigungsrelation für die Ableitung meromorpher Funktionen , 1971 .

[64]  S. Favorov.  Sunyer-i-Balaguer’s almost elliptic functions and Yosida’s normal functions , 2008, 0802.1487.

[65]  G. G. Gundersen,et al.  Meromorphic functions that share pairs of values , 1997 .

[66]  K. Yosida.  A note on malmquist's theorem on first order algebraic differential equations , 1977 .

[67]  S. Shimomura.  Poles and α-points of meromorphic solutions of the first Painlevé hierarchy , 2004 .

[68]  Alan F. Beardon,et al.  Iteration of Rational Functions: Complex Analytic Dynamical Systems , 1991 .

[69]  Shahar Nevo,et al.  Spherical derivatives and normal families , 2012 .

[70]  F. Brüggemann.  On the zeros of fundamental systems of linear differential equations with polynomial coefficients , 1990 .

[71]  H. Wittich.  Neuere Untersuchungen über eindeutige analytische Funktionen : mit 31 Abbildungen , 1968 .

[72]  RADIAL DISTRIBUTION AND DEFICIENCIES OF THE VALUES OF A MEROMORPHIC FUNCTION , 1961 .

[73]  Willibald A. Doeringer.  Exceptional values of differential polynomials. , 1982 .

[74]  G. Valiron.  Sur la dérivée des fonctions algébroïdes , 1931 .

[75]  R. Nevanlinna.  Eindeutige Analytische Funktionen , 1936 .

[76]  P. Lappan.  A criterion for a meromorphic function to be normal , 1974 .

[77]  F. Gross.  On the equation $f^n +g^n =1$. II , 1968 .

[78]  A. Hinkkanen,et al.  Growth results for Painlevé transcendents , 2004, Mathematical Proceedings of the Cambridge Philosophical Society.

[79]  Jianming Chang,et al.  Meromorphic Functions Sharing Four Values , 2003 .

[80]  L. Ahlfors,et al.  Complex Analysis Mcgraw Hill , 2014 .

[81]  N. Steinmetz.  A uniqueness theorem for three meromorphic functions , 1988 .

[82]  R. Redheffer,et al.  On the Growth of Logarithmic Derivatives , 1974 .

[83]  Zur Theorie linearer Differentialgleichungen im Komplexen , 1973 .

[84]  S. Wang.  On Meromorphic Functions That Share Four Values , 1993 .

[85]  R. Nevanlinna.  Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen , 1926 .

[86]  R. Kaufman,et al.  On Briot-Bouquet differential equations and a question of Einar Hille , 1981 .

[87]  A. Eremenko.  Meromorphic solutions of first-order algebraic differential equations , 1984 .

[88]  P. Painlevé.  Leçons sur la théorie analytique des équations différentielles : professées à Stockholm(1895) , 1897 .

[89]  George Polya,et al.  Aufgaben und Lehrsatze aus der Analysis , 1926 .

[90]  A. Eremenko.  On the Second Main Theorem of Cartan , 2014, 1409.4850.

[91]  N. Steinmetz.  Normal families and linear differential equations , 2012 .

[92]  Ilpo Laine,et al.  Nevanlinna Theory and Complex Differential Equations , 1992 .

[93]  T. Kecker.  Polynomial Hamiltonian systems with movable algebraic singularities , 2013, 1312.4030.

[94]  J. Clunie,et al.  The spherical derivative of integral and meromorphic functions , 1965 .

[95]  N. Steinmetz.  Meromorphic solutions of second-order algebraic differential equations , 1989 .

[96]  G. Hennekemper,et al.  Picardsche ausnahmewerte von ableitungen gewisser meromorpher funktionen , 1985 .

[97]  Rolf Newvanlinna.  Über Riemannsche Flächen mit endlich vielen Windungspunkten , 1932 .

[98]  E. Mues.  Über ein Problem von Hayman , 1979 .

[99]  G. Frank,et al.  Rational Deficient Functions of Meromorphic Functions , 1986 .

[100]  Margrit Frei.  Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten , 1961 .

[101]  Norbert Steinmetz.  Meromorphe lösungen der differentialgleichung , 1988 .

[102]  N. Steinmetz.  The Yosida class is universal , 2011, 1103.2073.

[103]  E. L. Ince.  Ordinary differential equations , 1927 .

[104]  M. Reinders.  A new example of meromorphic functions sharing four values and a uniqueness theorem , 1992 .

[105]  S. Bank.  A note on the zeros of solutions [wacute]+P(z)w= 0 where P is a polynomial , 1987 .

[106]  N. Steinmetz,et al.  The Theorem of Tumura‐Clunie for Meromorphic Functions , 1981 .

[107]  PARAMETRIZATIONS OF ALGEBRAIC CURVES , 2006 .

[108]  S. Bank.  A note on the location of complex zeros of solutions of linear differential equations , 1989 .

[109]  N. Steinmetz.  An old new class of meromorphic functions , 2016, 1601.04032.

[110]  A. Wiman.  Über den Zusammenhang Zwischen dem Maximalbetrage Einer Analytischen Funktion und dem Grössten Gliede der Zugehörigen Taylor'schen Reihe , 1914 .

[111]  Lawrence Zalcman,et al.  Normal families: New perspectives , 1998 .

[112]  Norbert Steinmetz.  Ein Malmquistscher Satz Für Algebraische Differentialgleichungen Zweiter Ordnung , 1986 .

[113]  W. Sternberg.  Über die asymptotische Integration von Differentialgleichungen , 1920 .

[114]  R. Kaufman,et al.  On meromorphic solutions of first-order differential equations , 1976 .

[115]  M. Reinders.  A New Characterization of Gundersen’s Example of Two Meromorphic Functions Sharing Four Values , 1993 .

[116]  N. Steinmetz.  A unified approach to the Painleve Transcendents , 2016, 1601.04031.

[117]  N. Steinmetz.  On the Zeros of a Certain Wronskian , 1988 .

[118]  N. Steinmetz.  On Painlevé's equations I, II and IV , 2000 .

[119]  W. Hayman,et al.  Picard Values of Meromorphic Functions and their Derivatives , 1959 .

[120]  K. Tohge,et al.  Meromorphic Functions that Share Four or Five Pairs of Values , 2018 .

[121]  G. Frank,et al.  On the Meromorphic Solutions of Non‐Homogeneous Linear Differential Equations with Polynomial Coefficients , 1986 .

[122]  Hideharu Ueda.  Some estimates for meromorphic functions sharing four values , 1994 .

[123]  W. Hayman,et al.  Research Problems in Complex Analysis , 1977 .

[124]  N. Steinmetz.  First‐order algebraic differential equations of genus zero , 2016, 1602.02512.

[125]  Alexandre Eremenko,et al.  Meromorphic solutions of algebraic differential equations , 1982 .

[126]  W. Bergweiler.  On a theorem of gol'dberg concerning meromorphic solutions of algebraic differential equations , 1998 .

[127]  G. G. Gundersen.  Meromorphic functions that share four values , 1983 .

[128]  A. Hinkkanen,et al.  Solutions of the first and second Painlevé equations are meromorphic , 1999 .

[129]  J. Clunie.  On Integral and Meromorphic Functions , 1962 .

[130]  E. Mues.  Meromorphic functions sharing four values , 1989 .

[131]  L. Rubel,et al.  Entire and Meromorphic Functions , 1996 .

[132]  J. Rossi,et al.  Zeros of meromorphic solutions of second order linear differential equations , 1986 .

[133]  A. Eremenko.  Normal holomorphic curves from parabolic regions to projective spaces Preprint , 2007, 0710.1281.

[134]  J. Langley.  Proof of a Conjecture of Hayman Concerning f and f , 1993 .

[135]  G. G. Gundersen.  Meromorphic Functions that Share Three or Four Values , 1979 .

[136]  V. Gavrilov.  SOME CLASSES OF MEROMORPHIC FUNCTIONS CHARACTERIZED BY THEIR SPHERICAL DERIVATIVE , 1968 .

[137]  L. Bieberbach.  Theorie der Gewöhnlichen Differentialgleichungen , 1953 .

[138]  Thomas Muir,et al.  A Treatise on the Theory of Determinants , 2015 .

[139]  F. Brüggemann.  On solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi , 1991 .

[140]  J. Malmquist,et al.  Sur les fonctions a un nombre fini de branches définies par les équations différentielles du premier ordre , 1913 .

[141]  The linear differential equation whose solutions are the products of solutions of two given differential equations , 1984 .

[142]  K. Yosida.  A Generalisation of a Malmquist's theorem , 1932 .

[143]  Günter Frank,et al.  Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen , 1976 .

[144]  Lawrence Zalcman,et al.  A Heuristic Principle in Complex Function Theory , 1975 .

[145]  P. Painlevé,et al.  Mémoire sur les équations différentielles dont l'intégrale générale est uniforme , 1900 .

[146]  K. Yosida.  On a Class of Meromorphic Functions , 1934 .

[147]  E. Hille,et al.  Ordinary di?erential equations in the complex domain , 1976 .

[148]  H. Ochiai.  Meromorphic Solutions of Some Difference Equations , 1982 .

[149]  F. Gross.  On the equation $f^n + g^n = 1$ , 1966 .

[150]  Katsutoshi Yamanoi,et al.  The second main theorem for small functions and related problems , 2004 .

[151]  G. G. Gundersen.  On the real zeros of solutions of f'' + A(z)f = 0 where A(z) is entire , 1986 .

[152]  Bloch’s Principle , 2005, math/0511048.

[153]  E. M. Steinbart,et al.  A generalization of the Airy integral for ”-ⁿ=0 , 1993 .

[154]  N. Steinmetz.  Über die Nullstellen von Differentialpolynomen , 1981 .

[155]  A. Gabrielov,et al.  SINGULAR PERTURBATION OF POLYNOMIAL POTENTIALS WITH APPLICATIONS TO PT-SYMMETRIC FAMILIES , 2010, 1005.1696.

[156]  R. Nevanlinna.  Zur Theorie der Meromorphen Funktionen , 1925 .

[157]  W. H. Young,et al.  Lectures On The General Theory Of Integral Functions , 1923 .

[158]  I. Ostrovskiǐ,et al.  Value Distribution of Meromorphic Functions , 2008 .