General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

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

https://www.math.purdue.edu/~eremenko/dvi/bergo.pdf

On the singularities of the inverse to a meromorphic function of finite order

The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography.

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Mathematics of thermoacoustic tomography

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.

/pdf/normal-families-new-perspectives-2jbc3t1pl3.pdf

Normal families: New perspectives

A Jacobi function ${\phi _\lambda }^{\left( {\alpha ,\beta } \right)}\left( {\alpha ,\beta ,\lambda \in C,\alpha \ne - 1, - 2,} \right)$ 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)

Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups

A Heuristic Principle in Complex Function Theory

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].

Normal Families and Shared Values

Analyticity and the Pompeiu problem

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.

Lawrence Zalcman

Papers

Normal families: New perspectives

A Heuristic Principle in Complex Function Theory

Normal Families and Shared Values

Analyticity and the Pompeiu problem

Analytic Capacity and Rational Approximation