Entire and meromorphic functions. With assistance from James E. Colliander
This book by the late Professor Rubel deals with many important topics from the theory of entire and meromorphic functions. There are 24 chapters, loosely connected, but the emphasis throughout is on the Nevanlinna theory with various applications. Frequently, the material is taken from earlier papers of the author and its collaborators (Taylor, Malliavin). -- Among the better known subjects are the following. Jensen's theorem, the first fundamental theorem and applications, Cartan's formulation of the characteristic and the Ahlfors-Shimizu characteristic. Borel's lemma follows, leading to estimates of \(\log M(r)\) via \(T(r)\) excluding sets of finite logarithmic length. Basic facts of the Wiman-Valiron theory (maximal term and index, Newton polygon) are next. Later, the second fundamental theorem and some applications are given includig functions that share distinct values. For example, if \(f\) is entire and \(f\) and \(f'\) share two distinct values, then \(f=f'\). Functions \(f\) of exponential type, their indicator diagram and its relation to the singularities of the Borel transform \(\phi\) of \(f\) are studied, and various generalizations of Carlson's identity theorem are given. -- Some less known topics are treated in a long chapter ``A Fourier series method''. Given \(f\) meromorphic in \(\mathbb{C}\), the Fourier coefficients \[ c_k(r,f)= {1\over 2\pi} \int^{2\pi}_0[\log|f(re^{\text{it}})|]e^{-ikt}dt \] are associated, and their growth is connected with the growth of \(f\). If \(\lambda(r)\nearrow\), the class \(\Lambda\) of functions \(f\) meromorphic in \(\mathbb{C}\) with \(T(r,f)\leq A\lambda(Br)\) (with constants \(A\) and \(B\)) and the class \(\Lambda_E\) of entire functions \(f\) with \(\log M(r)\leq A\lambda(Br)\) are studied including their zero sets and pole sets. It is shown that each meromorphic function \(f\) is \(f_1/f_2\) where \(T(r,f_k)\leq AT(Br,f)\) for \(k=1,2\). As an application of the Nevanlinna theory a lemma of Hiromi and Ozawa is included and later applied: If \(\sum^n_{k=1} a_k(z)e^{g_k(z)}=a_0(z)\) holds in \(\mathbb{C}\), for meromorphic \(a_k\) and entire \(g_k\), then there is an identity \(\sum^n_{k=1} c_ka_k(z)e^{g_k(z)}=0\) with \(c_k\) not all \(=0\), provided that the \(a_k\) are `small' in a certain sense. Noteworthy is also a chapter on integer-valued entire functions of exponential type (Theorem of Buck), and another theorem of Buck characterizing those \(\{A_n\}\) for which there exists an entire function \(f\) of exponential type with \(h_f(\pm{\pi\over 2})<\pi\) and \(f(n)=A_n\) \((n=1,2,\dots)\). The book closes with two chapters connecting to mathematical logic. For instance, if an expression \(t\) is made up of \(n\) variables and addition, multiplication, and exponentiation, and if \(t(z)=0\) for all \(z\in \mathbb{C}^n\), then there is an identity \(t=0\) starting from these basic operations. As is seen from the above, the book is very rich in content and treats diverse topics. However, the reviewer noticed that the different chapters have not been written with equal care. Sometimes an introduction, motivation and references to related work are missing. Often, more careful proof-reading would have been helpful, and in some places mathematical precision is lacking. Altogether, the book is a valuable addition to the literature on classical function theory.
scientific article; zbMATH DE number 877625
scientific article