Complex analysis is found in many areas of applied mathematics, from fluid mechanics, thermodynamics, signal processing, control theory, mechanical and electrical engineering to quantum mechanics, among others. And of course, it is a fundamental branch of pure mathematics. The coverage in this text includes advanced topics that are not always considered in more elementary texts. These topics include, a detailed treatment of univalent functions, harmonic functions, subharmonic and superharmonic functions, Nevanlinna theory, normal families, hyperbolic geometry, iteration of rational functions, and analytic number theory. As well, the text includes in depth discussions of the Dirichlet Problem, Green's function, Riemann Hypothesis, and the Laplace transform. Some beautiful color illustrations supplement the text of this most elegant subject.

The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. With its success, however, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. Even proofs of theorems often lack rigor, and dubious mathematical practices are not uncommon in the literature for students. In the present text, I have tried to bring to the subject a certain amount of mathematical correctness and make it accessible to un dergraduates. Th this end, this text addresses a number of issues that are rarely considered. For instance, when we apply the Laplace trans form method to a linear ordinary differential equation with constant coefficients, any(n) + an-lY(n-l) + . . . + aoy = f(t), why is it justified to take the Laplace transform of both sides of the equation (Theorem A. 6)? Or, in many proofs it is required to take the limit inside an integral. This is always fraught with danger, especially with an improper integral, and not always justified. I have given complete details (sometimes in the Appendix) whenever this procedure is required. IX X Preface Furthermore, it is sometimes desirable to take the Laplace trans form of an infinite series term by term. Again it is shown that this cannot always be done, and specific sufficient conditions are established to justify this operation."

Prior to the 1920s it was generally thought, with a few exceptions, that our galaxy, the Milky Way, was the entire Universe. Based on the work of Henrietta Leavitt with Cepheid variables, astronomer Edwin Hubble was able to determine that the Andromeda Galaxy and others had to lie outside our own. Moreover, based on the work of Vesto Slipher, involving the redshifts of these galaxies, Hubble was able to determine that the Universe was not static, as had been previously thought, but expanding. The number of galaxies has also been expanding, with estimates varying from 100 billion to 2 trillion. While every galaxy in the Universe is interesting just by its very fact of being, the author has selected 60 of those that possess some unusual qualities that make them of some particular interest. These galaxies have complex evolutionary histories, with some having supermassive black holes at their core, others are powerful radio sources, a very few are relatively nearby and even visible to the naked eye, whereas the light from one recent discovery has been travelling for the past 13.4 billion years to show us its infancy, and from a time when the Universe was in its infancy. And in spite of the vastness of the Universe, some galaxies are colliding with others, embraced in a graceful gravitational dance. Indeed, as the Andromeda Galaxy is heading towards us, a similar fate awaits our Milky Way. When looking at a modern image of a galaxy, one is in awe at the shear wondrous nature of such a magnificent creation, with its boundless secrets that it is keeping from us, its endless possibilities for harboring alien civilizations, and we remain left with the ultimate knowledge that we are connected to its glory.

The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. With its success, however, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. Even proofs of theorems often lack rigor, and dubious mathematical practices are not uncommon in the literature for students. In the present text, I have tried to bring to the subject a certain amount of mathematical correctness and make it accessible to un dergraduates. Th this end, this text addresses a number of issues that are rarely considered. For instance, when we apply the Laplace trans form method to a linear ordinary differential equation with constant coefficients, any(n) + an-lY(n-l) + * * * + aoy = f(t), why is it justified to take the Laplace transform of both sides of the equation (Theorem A. 6)? Or, in many proofs it is required to take the limit inside an integral. This is always fraught with danger, especially with an improper integral, and not always justified. I have given complete details (sometimes in the Appendix) whenever this procedure is required. IX X Preface Furthermore, it is sometimes desirable to take the Laplace trans form of an infinite series term by term. Again it is shown that this cannot always be done, and specific sufficient conditions are established to justify this operation.

A book on the subject of normal families more than sixty years after the publication of Montel's treatise Ler;ons sur les familles normales de fonc tions analytiques et leurs applications is certainly long overdue. But, in a sense, it is almost premature, as so much contemporary work is still being produced. To misquote Dickens, this is the best of times, this is the worst of times. The intervening years have seen developments on a broad front, many of which are taken up in this volume. A unified treatment of the classical theory is also presented, with some attempt made to preserve its classical flavour. Since its inception early this century the notion of a normal family has played a central role in the development of complex function theory. In fact, it is a concept lying at the very heart of the subject, weaving a line of thought through Picard's theorems, Schottky's theorem, and the Riemann mapping theorem, to many modern results on meromorphic functions via the Bloch principle. It is this latter that has provided considerable impetus over the years to the study of normal families, and continues to serve as a guiding hand to future work. Basically, it asserts that a family of analytic (meromorphic) functions defined by a particular property, P, is likely to be a normal family if an entire (meromorphic in"

I first had a quick look, then I started reading it. I couldn't stop. -Gerard 't Hooft (Nobel Prize, in Physics 1999) This is a book about the mathematical nature of our Universe. Armed with no more than basic high school mathematics, Dr. Joel L. Schiff takes you on a foray through some of the most intriguing aspects of the world around us. Along the way, you will visit the bizarre world of subatomic particles, honey bees and ants, galaxies, black holes, infinity, and more. Included are such goodies as measuring the speed of light with your microwave oven, determining the size of the Earth with a stick in the ground and the age of the Solar System from meteorites, understanding how the Theory of Relativity makes your everyday GPS system possible, and so much more. These topics are easily accessible to anyone who has ever brushed up against the Pythagorean Theorem and the symbol , with the lightest dusting of algebra. Through this book, science-curious readers will come to appreciate the patterns, seeming contradictions, and extraordinary mathematical beauty of our Universe.

Prior to the 1920s it was generally thought, with a few exceptions, that our galaxy, the Milky Way, was the entire Universe. Based on the work of Henrietta Leavitt with Cepheid variables, astronomer Edwin Hubble was able to determine that the Andromeda Galaxy and others had to lie outside our own. Moreover, based on the work of Vesto Slipher, involving the redshifts of these galaxies, Hubble was able to determine that the Universe was not static, as had been previously thought, but expanding. The number of galaxies has also been expanding, with estimates varying from 100 billion to 2 trillion. While every galaxy in the Universe is interesting just by its very fact of being, the author has selected 60 of those that possess some unusual qualities that make them of some particular interest. These galaxies have complex evolutionary histories, with some having supermassive black holes at their core, others are powerful radio sources, a very few are relatively nearby and even visible to the naked eye, whereas the light from one recent discovery has been travelling for the past 13.4 billion years to show us its infancy, and from a time when the Universe was in its infancy. And in spite of the vastness of the Universe, some galaxies are colliding with others, embraced in a graceful gravitational dance. Indeed, as the Andromeda Galaxy is heading towards us, a similar fate awaits our Milky Way. When looking at a modern image of a galaxy, one is in awe at the shear wondrous nature of such a magnificent creation, with its boundless secrets that it is keeping from us, its endless possibilities for harboring alien civilizations, and we remain left with the ultimate knowledge that we are connected to its glory.

Prior to the 1920s it was generally thought, with a few exceptions, that our galaxy, the Milky Way, was the entire universe. Based on the work of Henrietta Leavitt with Cepheid variables, astronomer Edwin Hubble was able to determine that others had to lie outside our own. This books looks at 60 of those that possess some unusual qualities that make them of particular interest, from supermassive black holes and colliding galaxies to powerful radio sources.

Dunedin-born artist Grace Joel (1864-1924) exhibited to acclaim in London and Paris, yet she and her art are relatively unknown today. Joel excelled at portraiture and mother and child studies, and was skilled in portraying the nude. She received her artistic training in Melbourne, and lived for the mature years of her career in London, where her work appeared at the prestigious Royal Academy, as well as the Paris Salon and the Royal Scottish Academy. She also held a number of solo exhibitions at prominent venues in Australasian, English and European cities. Today she is claimed by New Zealand, Australia and Britain. One possible reason why Joel's work has not remained visible is that few details of her personal life survive. Only three letters have been found, and they reveal little of the person who wrote them. Undaunted, author Joel (no relation) Schiff has pulled together from the words of her contemporaries, various newspaper accounts, scraps in other historical archives and close study of her extant paintings a portrayal of this talented woman that is as intimate and engaging as her work. He also sets Grace Joel and her work in the times in which she lived, and the artistic communities of which she was a part. Joel L. Schiff was born in Chicago, Illinois, USA and has a PhD in mathematics from the University of California, Los Angeles. He spent his career at the University of Auckland and written three books on various mathematical subjects. He has been a successful part-time asteroid hunter and is the founder/publisher of an international journal on meteorites. Dr Schiff's lifelong interest in art has led to him taking up the brush himself in recent years. He first encountered the work of Grace Joel 1981 at the Auckland City Art Gallery. The interest sparked then never died, and grew to an obsession over the decades. This book on her life and work is the result.