UNDERSTANDING MODERN MAGNETS THROUGH CONFORMAL MAPPING

When I had to choose, within some narrow range, the topic of this paper, I received great help from a colleague in Berkeley and from Prof. Little when it was suggested that I should pick among the possible subjects of my talk the subject that Prof. Bloch would have enjoyed most. Since Prof. Bloch would prefer a scalpel over a sword every time, I hope and think that most people will approve my choice. When one intends to talk about a subject that is as old as conformal mapping and one does not want to lose the audience in a very short time, it is advisable to start by explaining both the motivation for the talk as well as the goals one has in mind when giving the talk. This particular talk has been motivated by the increasing frequency with which one hears, from people that ought to know better, statements like: 'Conformal mapping is really a thing of the past because of all the marvelous computer programs that we now have'. Even though, or more likely because, I have been intimately involved in the development of some large and widely used computer codes, I am deeply disturbed by such more » statements since they indicate a severe lack of understanding of the purpose of conformal mapping techniques, computers, and computer codes. In my view, conformal mapping can be an extremely powerful computational technique, and the easy availability of computers has made that aspect even more important now than it has been in the past. Additionally, and more importantly, conformal mapping can give very deep and unique insight into problems, giving often solutions to problems that can not be obtained with any other method, in particular not with computers. Wanting to demonstrate in particular the latter part, I set myself two goals for this talk: (1) I want to show with the help of a number of examples that conformal mapping is a unique and enormously powerful tool for thinking about, and solving, problems. Usually one has to write down only a few equations, and sometimes none at all. When I started getting involved in work for which conformal mapping seemed to be a powerful tool, I did not think that I would ever be able to use that technique successfully because it seemed to require a nearly encyclopedic memory, an impression that was strengthened when I saw H.Kober's Dictionary of Conformal Representations (ref. 1). This attitude changed when I started to realize that beyond the basics of the theory of a function of a complex variable, I needed to know only about a handful of conformal maps and procedures. Consequently, my second goal for this talk is to: (2) Show that in most cases conformal mapping functions can be obtained by formulating the underlying physics appropriately. This means particularly that encyclopedic knowledge of conformal maps is not necessary for successful use of conformal mapping techniques. To demonstrate these facts I have chosen examples from an area of physics/engineering in which I am active, namely accelerator physics. In order to do that successfully I start with a brief introduction into high energy charged particle storage ring technology, even though not all examples used in this paper to elucidate my points come directly from this particular field of accelerator technology. This is followed by a brief summary of the most important properties of functions of a complex variable. When reading this introduction into the relevant mathematics, the reader needs to keep in mind that this is not a mathematics essay, but a demonstration how beautiful and powerful, but not always appreciated, mathematics can be if used by a physicist or engineer to solve some real life problems. « less