Riemann surfaces and asymptotic values associated with real entire functions.

  • 96 Pages
  • 0.92 MB
  • English
Rice Institute] , [Houston, Tex
Riemann surfaces., Asymptotes., Functions, En
SeriesRice Institute pamphlet. Special issue. Monograph in mathematics, Rice Institute pamphlet -- Nov. 1952., Rice Institute pamphlet
LC ClassificationsQA333 .M3
The Physical Object
Pagination96 p.
ID Numbers
Open LibraryOL22096389M
LC Control Number55038539

Riemann surfaces and asymptotic values associated with real entire functions. [Houston, Tex.], [Rice Institute] (OCoLC) Document Type: Book: All Authors / Contributors: Gerald R MacLane.

Riemann Surfaces and Asymptotic Values Associated with Real Entire Functions. By G. MacLane. Abstract. Preliminaries-- Caratheodory Kernels-- Real Entire Functions and Riemann Surfaces-- Reference Publisher: Rice University.

Year: OAI identifier Author: G. MacLane. [1] G. MacLane, Riemann surfaces and asymptotic values associated with real entire functions, The Rice Institute Pamphlet (). Mathematical Reviews (MathSciNet): MRe Zentralblatt MATH: Author: Howard B. Curtis. Another way to get the surface without gluing is to consider the sheaf of germs of analytic functions and take the connected component of one germ.

That is however again not very concrete. $\endgroup$ – Daniel Fischer ♦ May 10 '14 at Recall that a Riemann surface X is a connected one-dimensional complex manifold, i.e., a topological surface endowed with a conformal structure. A Riemann surface X is called analytically finite, or, more precisely, of analytic type (g, n, m), if it is conformally equivalent to a closed Riemann surface X 0 of genus g with n punctures and m holes bounded by analytic.

(Indeed, if a Riemann surface is embedded in Euclidean space, the co- ordinate functions of the ambient space restrict to holomorphic functions on the surface, and since the surface is compact without boundary, the real parts Riemann surfaces and asymptotic values associated with real entire functions.

book well as the moduli) of these restricted coordinate functions have interior local Size: KB.

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An asymptotic value of an entire or meromorphic function is a complex number a such that, as |z| → ∞ along a specified path, f(z) → a.

In this chapter, we consider two famous results involving such considerations. functions for the Laplace operator on Riemann surfaces. The Green function on a Riemann surface is an integral kernel which solves the Poisson equation 2i∂∂f¯ = φ.

More precisely, let Xbe a compact connected Riemann surface, and let µbe a smooth differential 2-form on Xsatisfying R X µ= 1.

Description Riemann surfaces and asymptotic values associated with real entire functions. FB2

Everything about Riemann surfaces. (absolute value of the) function must achieve a maximum, the maximum modulus principle from complex analysis will then tell you that this function is constant on an open set, and using the identity theorem on overlapping charts we find that the function is constant everywhere.

Miranda's book is as a. The point of the introduction of Riemann surfaces made by Riemann, Klein and Weyl (), was that Riemann surfaces can be considered as both a one-dimensional complex manifold and an algebraic curve.

Another possibility is to study Riemann surfaces as two-dimensional real manifolds, as Gauss () had taken on the problem of taking a piece of aFile Size: KB.

A value is a singular value if and only if it is an asymptotic value or a critical value. We show the construction of the parabolic simply connected Riemann surface associated with a fixed meromorphic function, and point out that every boundary point of the Riemann surface is an asymptotic value of the : Jianhua Zheng.

half{plane free of zeros are identi ed by the Riemann hypothesis for Hilbert spaces of entire functions [3]. The application of these spaces to the Riemann hypothesis for a large class of zeta functions is made possible by their construction in Fourier analysis [4]. The Riemann hypothesis is a conjecture concerning the zeros of a special entire function, not a.

This book establishes the basic function theory and complex geometry of Riemann surfaces, both open and compact. Many of the methods used in the book are adaptations and simplifications of methods from the theories of several complex variables and complex analytic geometry and would serve as excellent training for mathematicians wanting to work in complex Cited by: Riemann surface for the function f(z) = √z.

The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √z. The imaginary part of √z is represented by the coloration of the points. On the Riemann surface generated by a function meromorphic in the unit disk with ∞ as a spiral asymptotic value J.

Stebbins 1, 2 Mathematische Zeitschrift vol pages – () Cite this articleAuthor: J. Stebbins, J. Stebbins. This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. q >_ 2 degenerating to a Riemann surface of genus q- 1 with a non-separating node.

We show that the Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces simply degenerate to that on the smooth part of the noded Riemann surface.

Introduction. A Riemann–Hilbert approach to asymptotic questions for orthogonal polynomials Article in Journal of Computational and Applied Mathematics () August with 24 Reads. Thus, to any analytic function corresponds a Riemann surface on which this function is a single-valued analytic function of a point.

This means that in a neighbourhood of any point there exists a local uniformizing parameter in which is represented as a single-valued analytic function.

Geometric Theory of Functions of a Complex Variable. 5 On the number of asymptotic values of entire functions of finite order obtain obviously plane point ZQ polynomial positive number prove range of values real axis regular and univalent regular function result Riemann surface satisfy the condition Schwarz lemma segment set of points.

American Mathematical Society Charles Street Providence, Rhode Island or AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and services marks of the American Mathematical Society and registered in the U.S.

Patent and Trademark Office. In general, the general statement is that given Y a Riemann surface and some polynomial P (T) ∈ M (Y) [T] of degree n, one can find another Riemann surface X such that π: X ⟶ Y is an holomorphic n -branched covering and a meromorphic function F ∈.

I have always felt this was due to Riemann himself: especially in: Theory of Abelian Functions, Of course the association of a Riemann surface to an algebraic curve is generally attributed to him, and there he also proves conversely there is a plane curve associated to a (compact) Riemann surface.

Math. (), [3] C. Carathoodory, Untersuchungen ber die konformen Abbildungen von festen und vernderlichen Gebieten, Mathematische Annalen 72 (), [4] G. MacLane, Riemann surfaces and asymptotic values associated with real entire functions, Rice Institute Pamphlet, special issue, November [5] G.

real imaginary part of s, respectively; com-mon notation is σ= Res and t = Ims. t,x,y Real variables. Res s=s F(s) The residue of F(s) at the point s = s. ζ(s) Riemann’s zeta-function defined by ζ(s) = X∞ n=1 n−s for Res > 1 and for other values of s by analytic continuation.

γ(s) The gamma-function, defined for Res >0 by Γ(s) = Z File Size: 2MB. Asymptotic characteristics of entire functions and their applications in mathematics and biophysics. [L S Maergoĭz] --Appendix A Riemann Surface of the Inverse Function for a Polynomial of Fractional Order --A.1 Some functions --B.3 Entire functions associated with an analytic proximate order --B.4 Entire.

The book provides a basic introduction to the development of the theory of entire and meromorphic functions from the s to the early s. After an opening chapter introducing fundamentals of Nevanlinna's value distribution theory, this book discusses various relationships among and developments of three central concepts: deficient value.

forthcoming book on the trace formula. We will illustrate the method in the case of a compact Riemann surface, and begin by recalling a few facts about the Selberg zeta-function Z(s) associated with the surface [4], [7], [10]. To begin with, Z(s), for Re s > 1, is defined by the product 00 I I (1-e-4(s+n) Y n-O.

A Riemann Surface X is called hyperlliptic if it admits a 2 to 1 holomorphic map φ:X → P 1. Given a (compact) Riemann Surface X of genus g ≥3 which is not hyperelliptic one can define an embedding φ K:X → P g-1 called the canonical embedding (the construction of this map can found in Rick Miranda's book "Algebraic Curves and Riemann.

Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations.

The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. Thus, Riemann surfaces appear as the unramified or ramified surfaces associated to an analytic germ.

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One highlight of this chapter is the construction of the complex projective algebraic curve associated to a compact Riemann surface. This construction is illustrated with several examples, including the case of elliptic curves.Goals of the Lecture: To see how the real plane can be equipped with two different Riemann Surface structures - To see how the real 2-dimensional sphere can be equipped with a Riemann surface structure.

Topics: Complex coordinate chart, compatible charts, transition function, complex atlas, Riemann surface, holomorphic function, unit disc.The Riemann Hypothesis: Arithmetic and Geometry Jeffrey C. Lagarias (May 4, ) ABSTRACT This paper describes basic properties of the Riemann zeta function and its generalizations, indicates some of geometric analogies, and presents various for-mulations of the Riemann hypothesis.

It briefly discusses the approach of Size: KB.