Analytic Functions

Analytic Functions

Copyright Date: 1960
Pages: 206
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    Analytic Functions
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    eISBN: 978-1-4008-7670-9
    Subjects: Mathematics
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Table of Contents

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  1. Front Matter (pp. i-iv)
  2. Preface (pp. v-vi)
    Marston Morse, Arne Beurling and Atle Selberg
  3. Table of Contents (pp. vii-2)
  4. On Differentiable Mappings (pp. 3-10)
    Rolf Nevanlinna

    The treatment of the problem of mappings in linear spaces can be based formally as well as intuitively on an infinitesimal calculus which is coordinate-free and dimension-free. Such a differential calculus has been introduced by Fréchet for infinite-dimensional spaces, and has been applied by him and others after him (Graves, Hildebrandt, Taylor, Rothe, Lusternik, Sobolev, etc.), principally in functional analysis. In the course of the last five years my brother and I have developed systematically such an infinitesimal technique in lectures in differential geometry, in the theory of manifolds and partial differential equations, and in the calculus of variations [1]....

  5. Analysis in Non-Compact Complex Spaces (pp. 11-44)
    H. Behnke and H. Grauert

    About one hundred years have elapsed since the recognition that as domains of existence of analytic functions of a complex variable, the Riemann surfaces appear. The analogue of Riemann surfaces for analytic functions of several complex variables—that is, the complex spaces—has, however, become known only in the past few years. This, of course, is not accidental. Already the first specific results in the field of function theory of several complex variables, discovered by Henri Poincaré, Pierre Cousin, and Fritz Hartogs at the turn of the century, showed that a function theory of several complex variables must necessarily be...

  6. The Complex Analytic Structure of the Space of Closed Riemann Surfaces (pp. 45-66)
    Lars V. Ahlfors

    In the classical theory of algebraic curves many attempts were made to determine the “modules” of an algebraic curve. The problem was vaguely formulated, and the only tangible result was that the classes of birationally equivalent algebraic curves of genusg> 1 depend on 6g−6 real parameters. More recent attempts to go to the bottom of the problem by more powerful algebraic methods have also ended in failure.

    The corresponding transcendental problem is to study the space of closed Riemann surfaces and, if possible, introduce a complex analytic structure on that space. In this direction considerable progress has been made....

  7. Some Remarks on Perturbation of Structure (pp. 67-88)
    Donald C. Spencer

    The basic technique used by Kodaira and Spencer in their paper [6] to describe the deformation of complex analytic structure is applied to other structures, in particular to real foliate structure (see, e.g., Reeb [7]) and to structures (real and complex) which are defined by simple infinite pseudo-groups in the sense of E. Cartan [1]. Comparison is made with the treatment of complex analytic structure described in [6]; this provides insight into the mechanism used and elucidates the rôle played by the theory of harmonic forms.

    A systematic treatment of the deformation of real and complex multifoliate structures will be...

  8. Quasiconformal Mappings and Teichmüller’s Theorem (pp. 89-120)
    Lipman Bers

    LetSbe a Riemann surface andfa homeomorphism ofSonto another Riemann surfaceS′. Iffis, in terms of local parameters, of classC1and has a positive Jacobian, then the deviation of this mapping from conformality can be measured, at each pointpofS, by the ratioKp> 1 of the axes of the infinitesimal ellipse into whichftakes an infinitesimal circle located atp. We setK[f] = supKp. The “overall dilatation”K[f] can be defined also for a somewhat wider class of “quasiconformal” homeomorphisms.

    Teichmuller’s theorem asserts that given any...

  9. On Compact Analytic Surfaces (pp. 121-136)
    Kunihiko Kodaira

    The present note is a preliminary report on a study of structures of compact analytic surfaces.

    1. LetVbe a compact analytic surface, i.e. a compact complex manifold of complex dimension 2. LetM(V) be the field of all meromorphic functions onVand denote by dimM(V) the degree of transcendency ofM(V) over the fieldCof all complex numbers. By a result due to Chow [2] and Siegel [9], we have

    dimM(V) ≦ 2.

    We denote bypg(V) the geometric genus ofV, byc1the first Chern class ofVand by$c_{1}^{2}(V)$the value...

  10. The Conformal Mapping of Riemann Surfaces (pp. 137-158)
    Maurice Heins

    The object of this essay is to examine the status of the chapter of the theory of Riemann surfaces which is concerned with conformal maps of Riemann surfaces. It is to be understood that we take the term in its general sense and do not require the considered maps to beunivalent onto maps. We set as our goal a study of the following problems: (1) the relation between conformal mapping theory so conceived and other chapters of the general theory of Riemann surfaces, (2) the development of an autonomous mapping theory in which conformal maps are studied for their...

  11. On Certain Coefficients of Univalent Functions (pp. 159-194)
    James A. Jenkins

    1. Despite the passage of years, Teichmüller’s coefficient theorem [15] remains the most penetrating explicit result known in the general coefficient problem for univalent functions. It is strange, then, that these considerations have up to the present been of little use in finding explicit numerical bounds for the coefficients of lower order. The principal purpose of the present account is to show how a closely related method proves exceedingly effective in finding such bounds.

    2. If ℛ is a Riemann surface, by a quadratic differential defined on 91 we mean an entity which assigns to every local uniformizing parameter of ℛ and...

  12. Appendix: Papers presented in the Seminars (Titles and Authors) (pp. 195-197)


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