In preparing the English edition of this unique work, every effort has been made to obtain an easily read and lueid exposition of the material. This has frequently been done at the expense of a literal translation of the original text and it is felt that such liberties as have been taken with the author's language are justified in the interest of ease in readingo None of us pretends to be an authority in the Russian language, and we trust that the original intent of the authors has not been lost. The equations, whieh were for the most part taken verbatim from the original work, were eheeked only eursorily; obvious and previously noted errors have been eorreeted. Fortunately, the Russian and English mathematieal notations are generally in good agreement. An exeeption is the shortened abbreviations for the hyperbolie functions (e.g. sh for sinh), and the symbol Jm rather that Im to denote the imaginary part. As near as possible, these diserepaneies have been correeted. In preparing the Bibliography, works having an English equivalent have been translated into the English title, but in the text the referenee to the Russian work was retained, as it was impraetieal to attempt to find in eaeh ease the eorresponding eitation in the English edition. Authors' names and titles associated with purely Russian works have been transliterated as nearly as possible to the English equivalent, along with the equivalent English title of the work cited.

Bessel functions are associated with a wide range of problems in important areas of mathematical physics. Bessel function theory is applied to problems of acoustics, radio physics, hydrodynamics, and atomic and nuclear physics. Bessel Functions and Their Applications consists of two parts. In Part One, the author presents a clear and rigorous intro

The subject of special functions is rich and expanding continuously with the emergence of new problems encountered in engineering and applied science applications. The development of computational techniques and the rapid growth in computing power have increased the importance of the special functions and their formulae for analytic representations

Around 1970, an abrupt change occurred in the study of holomorphic functions of several complex variables. Sheaves vanished into the back ground, and attention was focused on integral formulas and on the "hard analysis" problems that could be attacked with them: boundary behavior, complex-tangential phenomena, solutions of the J-problem with control over growth and smoothness, quantitative theorems about zero-varieties, and so on. The present book describes some of these developments in the simple setting of the unit ball of en. There are several reasons for choosing the ball for our principal stage. The ball is the prototype of two important classes of regions that have been studied in depth, namely the strictly pseudoconvex domains and the bounded symmetric ones. The presence of the second structure (i.e., the existence of a transitive group of automorphisms) makes it possible to develop the basic machinery with a minimum of fuss and bother. The principal ideas can be presented quite concretely and explicitly in the ball, and one can quickly arrive at specific theorems of obvious interest. Once one has seen these in this simple context, it should be much easier to learn the more complicated machinery (developed largely by Henkin and his co-workers) that extends them to arbitrary strictly pseudoconvex domains. In some parts of the book (for instance, in Chapters 14-16) it would, however, have been unnatural to confine our attention exclusively to the ball, and no significant simplifications would have resulted from such a restriction.

This book presents a collection of independent mathematical studies, describing the analytical reduction of complex generic problems in the theory of scattering and propagation of electromagnetic waves in the presence of imperfectly conducting objects. Their subjects include: a global method for scattering by a multimode plane; diffraction by an impedance curved wedge; scattering by impedance polygons; advanced properties of spectral functions in frequency and time domains; bianisotropic media and related coupling expressions; and exact and asymptotic reductions of surface radiation integrals. The methods developed here can be qualified as analytical when they lead to exact explicit expressions, or semi-analytical when they drastically reduce the mathematical complexity of studied problems. Therefore, they can be used in mathematical physics and engineering to analyse and model, but also in applied mathematics to calculate the scattered fields in electromagnetism for a low computational cost.

1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2) fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J am 1

A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems x'(t) = f(t, x(t), u(t)), x(O)=xo "controlled" by parameters u(t) (the "controls"). Indeed, if we introduce the set-valued map F(t, x)= {f(t, x, u)}ueu then solutions to the differential equations (*) are solutions to the "differen tial inclusion" (**) x'(t)EF(t, x(t)), x(O)=xo in which the controls do not appear explicitely. Systems Theory provides dynamical systems of the form d x'(t)=A(x(t)) dt (B(x(t))+ C(x(t)); x(O)=xo in which the velocity of the state of the system depends not only upon the x(t) of the system at time t, but also on variations of observations state B(x(t)) of the state. This is a particular case of an implicit differential equation f(t, x(t), x'(t)) = 0 which can be regarded as a differential inclusion (**), where the right-hand side F is defined by F(t, x)= {vlf(t, x, v)=O}. During the 60's and 70's, a special class of differential inclusions was thoroughly investigated: those of the form X'(t)E - A(x(t)), x (0) =xo where A is a "maximal monotone" map. This class of inclusions contains the class of "gradient inclusions" which generalize the usual gradient equations x'(t) = -VV(x(t)), x(O)=xo when V is a differentiable "potential". 2 Introduction There are many instances when potential functions are not differentiable

Bei der Herausgabe der KLEINschen Vorlesung über die hyper geometrische Funktion erschienen nur zwei Wege gangbar: Entweder eine durchgreifende Umarbeitung, auch im großen, oder eine möglichst weitgehende Erhaltung der ursprünglichen Form. Vor allem auch aus historischen Gründen wurde der letztere Weg beschritten. Daher ist die Anordnung des Stoffes erhalten geblieben; e, s ist nur, von kleinen Änderungen abgesehen, ein Exkurs über homogene Schreibweise aus der KLEINschen Vorlesung über lineare Differentialgleichungen ein gefügt, ferner sind die Schlußbemerkungen zur geometrischen Theorie im Falle komplexer Exponenten als durch die Arbeiten von F. SCHILLING überholt, weggelassen. Aus dem obengenannten Grunde sind beispiels weise auch Entwicklungen beibehalten worden, die heute schon dem Anfänger geläufig sind (etwa die Ausführungen über stereographische Projektion). In Rücksicht auf möglichste Erhaltung der KLEINschen Darstellung sind ferner Hinweise des Herausgebers auf inzwischen ge machte Fortschritte der Wissenschaft vom Texte getrennt als Anmerkun gen am Schluß zusammengestellt. Diese Hinweise erheben aber in keiner Weise den Anspruch auf Vollständigkeit. Bei der nicht zu um gehenden Revision des Textes im einzelnen ist, dem oben angegebenen Gesichtspunkt entsprechend, möglichste Wahrung des persönlichen KLEINschen Stils angestrebt. übrigens habe ich darauf Bedacht genommen, auch dem A nlänger die Lektüre durch Anmerkungen und durch Nachweise der KLEINschen Zitate zu erleichtern. Denn zweifellos bieten gerade diese Vorlesungen eine treffliche Ergänzung und Weiterführung dessen, was der Studierende mittleren Semesters an Geometrie und Funktionentheorie kennen gelernt hat.

In the present book, we have put together the basic theory of the units and cuspidal divisor class group in the modular function fields, developed over the past few years. Let i) be the upper half plane, and N a positive integer. Let r(N) be the subgroup of SL (Z) consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex analytic isomorphic to an affine curve YeN), whose compactifi cation is called the modular curve X(N). The affine ring of regular functions on yeN) over C is the integral closure of C[j] in the function field of X(N) over C. Here j is the classical modular function. However, for arithmetic applications, one considers the curve as defined over the cyclotomic field Q(JlN) of N-th roots of unity, and one takes the integral closure either of Q[j] or Z[j], depending on how much arithmetic one wants to throw in. The units in these rings consist of those modular functions which have no zeros or poles in the upper half plane. The points of X(N) which lie at infinity, that is which do not correspond to points on the above affine set, are called the cusps, because of the way they look in a fundamental domain in the upper half plane. They generate a subgroup of the divisor class group, which turns out to be finite, and is called the cuspidal divisor class group.