scispace - formally typeset
Search or ask a question
Author

Lawrence Zalcman

Bio: Lawrence Zalcman is an academic researcher from Bar-Ilan University. The author has contributed to research in topics: Meromorphic function & Holomorphic function. The author has an hindex of 21, co-authored 80 publications receiving 2302 citations. Previous affiliations of Lawrence Zalcman include Technion – Israel Institute of Technology & Stanford University.


Papers
More filters
Journal ArticleDOI
TL;DR: A survey of the application of the Bloch's Principle to one-variable theory can be found in this article, with the aim of making this technique available to as broad an audience as possible.
Abstract: This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle. Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch’s Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report. One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale. 1. Let D be a domain in the complex plane C. We shall be concerned with analytic maps (i.e., meromorphic functions) f : (D, | |R2) → (Ĉ, χ) Received by the editors October 15, 1997, and, in revised form, May 26, 1998. 1991 Mathematics Subject Classification. Primary 30D45; Secondary 30D35, 34A20, 58F23.

361 citations

Journal ArticleDOI
TL;DR: In this article, a family of meromorphic functions on the plane domain D and a ∈ Copf is defined as a family whose zeros are of multiplicity at least k. The case Ēf(0) = O is a celebrated result.
Abstract: For f a meromorphic function on the plane domain D and a ∈ [Copf ], let Ēf(a) = {z ∈ D[ratio ]f(z) = a}. Let [Fscr ] be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ [Fscr ], Ēf(0) = Ēf(k)(b) and 0 < [mid ]f(k+1)(z)[mid ] [les ] h whenever z ∈ Ēf(0), then [Fscr ] is a normal family on D. The case Ēf(0) = O is a celebrated result of Gu [5].

238 citations

Book
01 Jan 1968
TL;DR: In this paper, the problem of rational approximation of integrals has been studied in the context of function algebra and function algebra methods, and applications of Vitushkin's theorem have been discussed.
Abstract: Peak points.- Analytic capacity.- Some useful facts.- Estimates for integrals.- Melnikov's theorem.- Further results.- Applications.- The problem of rational approximation.- AC capacity.- A scheme for approximation.- Vitushkin's theorem.- Applications of Vitushkin's theorem.- Geometric conditions.- Function algebra methods.- Some open questions.

109 citations


Cited by
More filters
Journal ArticleDOI
TL;DR: In this paper, it was shown that if f is a transcendental meromorphic function, then f with n = 1 takes every finite non-zero value infinitely often, which is a conjecture of Hayman.
Abstract: Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ?, then every asymptotic value of f, except at most 2? of them, is a limit point of critical values of f. We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f'fn with n = 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions

389 citations

Journal ArticleDOI
TL;DR: A survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography can be found in this paper.
Abstract: The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography.

385 citations

Journal ArticleDOI
TL;DR: A survey of the application of the Bloch's Principle to one-variable theory can be found in this article, with the aim of making this technique available to as broad an audience as possible.
Abstract: This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle. Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch’s Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report. One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale. 1. Let D be a domain in the complex plane C. We shall be concerned with analytic maps (i.e., meromorphic functions) f : (D, | |R2) → (Ĉ, χ) Received by the editors October 15, 1997, and, in revised form, May 26, 1998. 1991 Mathematics Subject Classification. Primary 30D45; Secondary 30D35, 34A20, 58F23.

361 citations

Book ChapterDOI
01 Jan 1984
TL;DR: A Jacobi function is defined as a even C∞-function on ℝ which equals 1 at 0 and which satisfies the differential equation as mentioned in this paper, where the Jacobi functions are defined as functions that satisfy the even C ∞-approximation.
Abstract: A Jacobi function \({\phi _\lambda }^{\left( {\alpha ,\beta } \right)}\left( {\alpha ,\beta ,\lambda \in C,\alpha e - 1, - 2,} \right)\) is defined as the even C∞-function on ℝ which equals 1 at 0 and which satisfies the differential equation $$\left( {{d^2}/d{t^2} + \left( {\left( {2\alpha + 1} \right)cotht + \left( {2\beta + 1} \right)tht} \right)d/dt + + {\lambda ^2} + {{\left( {\alpha + \beta + 1} \right)}^2}} \right){f_\lambda }^{\left( {\alpha ,\beta } \right)}\left( t \right) = 0$$ (11)

342 citations