1.1. Contents

1. Differentiable functions in $\mathbb{R}^{n}$

1. Taylor's formula
2. Partitions of unity
3. Inverse functions, implicit functions and the rank theorem
4. Sard's theorem and functional dependence
5. E Borel's theorem and approximation theorems
6. Ordinary differential equations

2. Manifolds

1. Basic definitions
2. Vector fields and differential forms
3. Submanifolds
4. Exterior differentiation
5. Orientation and Integration
6. One parameter groups and the theorem of Frobenius
7. Poincare's lemma, the type decomposition
8. Applications to complex analysis
9. Immersions and imbeddings: the theorems of Whitney

3.

1. Vector bundles
2. Linear differential operators: the theorem of Peetre
3. The Cauchy Kovalevski Theorem
4. Fourier transforms, Plancherel's theorem
5. The Sobolev spaces $H_{m,p}$
6. Elliptic differential operators
7. Elliptic operators with $C^{∞}$
8. Elliptic operators with analytic coefficients
9. The finiteness theorem
10. The approximation theorem and its ...

1.2. Review by: J Eells.
Mathematical Reviews MR0212837 (35 #3702).

These are detailed notes of a course of lectures treating topics in analysis on manifolds. In particular, they include: (1) The implicit function theorem, Sard's theorem, Whitney's imbedding theorem, the exterior calculus, Frobenius' theorem; (2) Grothendieck's lemma, Poincaré lemma, Hartog's continuation theorem, the Cauchy-Kowalewski theorem; (3) Various smooth and holomorphic approximation and extension theorems; (4) Linear differential operators in vector bundles on manifolds, elliptic operators (using the techniques of Sobolev spaces and inequalities of Gårding and Friedrichs), analytic elliptic operators. Throughout there is a nice blend of the real and complex aspects of the theory - certainly a refreshing exposition. {The reviewer has been informed that these notes are the basis of a book now in preparation.}

1.3. Review by: M S Narasimhan.
zbMATH 0185.33601.

This is an excellent set of lecture notes covering some basic topics in real and complex analysis. The topics covered are of interest not only to students of analysis but also to a wider mathematical public.

The first chapter is concerned with differentiable functions on $\mathbb{R}^{n}$. Among the results proved are an implicit function theorem, Sard's theorem and the theorem on approximation by analytic functions. The second chapter deals with differential calculus on manifolds. One finds proofs of Stokes' theorem and the theorem of Frobenius on involutive differential systems. The Poincaré lemma for $d$ - and $\bar{\partial}$-operators is proved. This is applied to prove Hartogs' continuation theorem and the Oka-Weil theorem. A proof of Whitney's imbedding theorem is also given. The last chapter deals with linear elliptic operators acting on sections of vector bundles. Regularity theorems for elliptic operators with $C^{∞}$ and analytic coefficients are proved, using a-priori estimates. There is also a section on elliptic operators on compact manifolds. The last section deals with a Runge type approximation theorem for solutions of linear elliptic equations; as an application it is shown that an open Riemann surface is a Stein manifold.

2.1. From the Preface.

The aim of these notes is to give proofs of the basic theorems in the local theory of analytic spaces and a few of their applications to results on the global structure of complex analytic spaces. The classical theory of holomorphic and analytic functions in $\mathbb{C}^{n}$ and $\mathbb{R}^{n}$ respectively is assumed; specific references are given for the results that are assumed. The elements of commutative algebra, especially properties of Noetherian and factorial rings, and of elements integral over a ring, are also assumed.

The term "analytic space" does not occur in the text. In conformity with German practice, we have called analytic spaces over the complex numbers simply "complex spaces". Further, analytic spaces over the real numbers are not introduced in all generality. We have considered only analytic subsets of a real analytic manifold, since a satisfactory treatment of the general case would involve results of Cartan and Bruhat - Whitney which we have stated but not proved. Nevertheless it was felt that the present title was the most appropriate.

2.2. Contents.

Preface
Chapter I. Preliminaries
Chapter II. The Weierstrass preparation theorem
Chapter III. Local properties of analytic sets
Chapter IV. Coherence theorems
Chapter V. Real analytic sets
Chapter VI. The normalization theorem
Chapter VII. Holomorphic mappings of complex spaces
Bibliographical Notes
References

2.3. Review by: W Kaup.
Mathematical Reviews MR0217337 (36 #428).

The author provides an excellent introduction to the local theory of complex spaces, always including the real case where possible. This inexpensive booklet contains all the fundamental theorems of this theory along with complete proofs. Some knowledge of commutative algebra (factorial and Noetherian rings, etc.) and familiarity with holomorphic functions of several complex variables are assumed; otherwise, no special borrowings from other fields are made. Nevertheless, beginners will find it challenging to work through the text. The presentation is quite concise and demanding - moreover, (with the exception of Chapter 5) there are hardly any examples to illustrate the concepts.

Specifically, the book is divided into the following chapters: (1) Preliminaries (A brief overview of the classical theory of holomorphic functions of n variables is given). (2) The Weierstrass preparation theorem (Four different proofs are given for this important theorem, all of which are relevant to what follows. One of the proofs is based on ideas by Malgrange - another is unpublished and is by Grauert and Remmert). (3) Local properties of analytic sets (Relationship between analytic set seeds and ideals of analytic function seeds; considerations of dimensions; Hilbert's zero-intercept theorem; analyticity of the singularity set; completeness of the vector space of all holomorphic functions on an analytic set). (4) Coherence theorems (Definition of complex space; coherence of the structure sheaf of a complex space; coherence of zeroth image sheaves for compact discrete holomorphic mappings; meromorphic functions). (5) Real analytic sets (Presentation of the results of Bruhat and Cartan and Bruhat and Whitney, respectively, partly without proofs). (6) The normalisation theorem. (7) Holomorphic mappings of complex spaces (Remmert-Stein theorem on the continuation of analytic sets; Chow theorem; analyticity of the image under closed holomorphic mappings; continuation of meromorphic functions; algebraic and analytic dependence of functions). Bibliographical notes.

2.4. Review by: Michel Hervé.
zbMATH 0168.06003.

This course is almost entirely devoted to complex analytic spaces. However, the case where the base field $K$ is that of the real numbers is considered several times:$K = \mathbb{R}$ or $\mathbb{C}$ is taken interchangeably in the exposition of the preparatory theorems, the fundamental properties of the ring of germs of analytic functions at a given point, in the construction of a basis of $K^{n}$ suitable for the local study of a given irreducible germ of an analytic set, and in the definition of its dimension.

On the other hand, Chapter V pushes the study of real analytic sets quite far: it establishes the main results of Bruhat-Cartan on the irreducible components of these sets, gives without proof some results of Bruhat-Whitney on C-analytic sets, and ends with examples, due to Bruhat and Cartan, which highlight the differences with the complex case.

The first 3 chapters contain what one needs to know of local theory to approach the study of complex analytic spaces: in these 3 chapters, the rapporteur has the honour of seeing taken up again the methods of local theory that he proposed in 1962, in a book published in Bombay [Several complex variables. Local theory (1963)], and the satisfaction of seeing its presentation improved on several points.

Chapter IV, devoted to consistency theorems, assumes knowledge of the first chapter of J-P Serre's Mémoire on coherent algebraic sheaves (1955)]: this is, moreover, apart from the basic notions, all that the author assumes to be known. He introduces complex analytic spaces at the beginning of this Chapter IV; then he shows, following Oka, that the sheaf of germs of holomorphic functions on an open set of $\mathbb{C}^{n}$ is coherent, i.e., that the sheaf of relations is of finite type; Following Cartan, that the sheaf of ideals of an analytic set is coherent, i.e., of finite type, he finally gives a new proof of the Grauert-Remmert theorem according to which, if $X$ and $Y$ are two complex analytic spaces and f a proper holomorphic map, with discrete fibres, from $X$ to $Y$, then the direct image under $f$ of a coherent sheaf of $X$-modules is a coherent sheaf of $Y$-modules.

Chapter VI introduces the notion of a weakly holomorphic function (i.e., holomorphic on the set of regular points and bounded in the neighbourhood of every singular point) on a complex analytic space, then that of a normal space; He demonstrates the theorem, due to Oka, of the existence of a normalisation of every complex analytic space $X$, and deduces from it the consistency of the sheaf of weakly holomorphic function seeds on $X$; he concludes with a new proof, due to Grauert-Remmert, of the analyticity of the set of points where $X$ is not normal.

Chapter VII begins by demonstrating a theorem of Remmert-Stein on the extension of an analytic subset of $X-Y$ (where $Y$ is an analytic set in the space $X$) to an analytic subset of $X$; From this, he deduces, firstly, Chow's theorem on analytic sets in a complex projective space, and secondly, Remmert's theorem, according to which, if $X$ and $Y$ are two complex analytic spaces, the image of a proper (or closed, as the author shows) holomorphic map is analytic in $Y$. From this theorem, he deduces, firstly, Thimm's theorem linking algebraic and analytic dependence on a compact complex space, and secondly, on a normal complex space, Levi's extension theorem for meromorphic functions, even improving the proofs found in the literature for the classical case of an open set of $\mathbb{C}^{n}$. The rapporteur hopes to have shown, through the preceding summary, that this course has the great merit of making the statements and proofs of several fundamental theorems much more accessible; this effectiveness is enhanced by the author's knowledge. of certain unpublished works, as well as by the clarity of the writing and the elegance of the presentation.

3.1. From the Publisher.

Chapter 1 presents theorems on differentiable functions often used in differential topology, such as the implicit function theorem, Sard's theorem and Whitney's approximation theorem.

The next chapter is an introduction to real and complex manifolds. It contains an exposition of the theorem of Frobenius, the lemmata of Poincaré and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thom's transversality theorem.

Chapter 3 includes characterisations of linear differentiable operators, due to Peetre and Hormander. The inequalities of Garding and of Friedrichs on elliptic operators are proved and are used to prove the regularity of weak solutions of elliptic equations. The chapter ends with the approximation theorem of Malgrange-Lax and its application to the proof of the Runge theorem on open Riemann surfaces due to Behnke and Stein.

3.2. From the Preface.

This book has its origin in lectures given at the Tata Institute of Fundamental Research, Bombay in the winter of 1964/65. The aim of the lectures was to present various topics in analysis, both on real and on complex manifolds. It is unnecessary to add that the topics actually chosen were determined entirely by personal taste. The contents were issued as lecture notes by the Tata Institute, and the present book is based on these notes.

The book is meant for people interested in analysis, who have little analytical background. The elements of the theory of functions of real variables (differential and integral calculus and measure theory) and some complex variable theory are assumed. Elementary properties of functions of several complex variables which are used are, in general, stated explicitly with references. It is however supposed that the reader is well acquainted with linear and multilinear algebra (properties of duals, tensor products, exterior products and so on of vector spaces) as well as set topology (properties of connected and locally compact spaces).

There are three chapters. The first deals with properties of differentiable functions in $\mathbb{R}^{n}$. The aim is to present, with complete proofs, some theorems on differentiable functions which are often used in differential topology (such as the implicit function theorem, Sard's theorem and Whitneys' approximation theorem).

The second chapter is meant as an introduction to the study of real and complex manifolds. Apart from the usual definitions (differential forms and vector fields) this chapter contains an exposition of the theorem of Frobenius, the lemmata of Poincard and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thorn's transversality theorem.

The last chapter deals with properties of linear elliptic differential operators. Characterisations of linear differential operators, due to Peetre and to Harmander are given. The inequalities of Girding and of Friedrichs on elliptic operators are proved and are used to. prove the regularity of weak solutions of elliptic equations. The chapter ends with the approximation theorem of Malgrange-Lax and its application to the proof of the Runge theorem on open Riemann surfaces due to Behnke and Stein.

We have not dealt with Riemannian metrics and elementary differential geometry. Nor have we dealt with elliptic complexes in spite of their importance and interest. It is actually not very difficult to extend the theorems, such as the finiteness theorem of Chapter 3, to such complexes.

3.3. Review by: J W Robbin.
Mathematical Reviews MR0251745 (40 #4972).

This book covers the basic material prerequisite to the study of differential equations (both ordinary and partial) on manifolds.

In Chapter I the local theory of differentiable functions is explained. The context is finite-dimensional, and the notation of multi-indices, rather than of multi-linear maps, is used for the higher derivatives. Taylor's formula, various versions of the implicit function theorem, $C^{∞}$ Sard's theorem, Borel's theorem (given a Taylor series there is a $C^{∞}$ function, etc.), Whitney's approximation theorem (a $\bar{\partial}$ function may be $C^{k}$ approximated by a real analytic), Runge's theorem, and the existence and uniqueness theorem for ordinary differential equations are proved.

Chapter II deals with manifolds, forms, vector fields, etc. The definitions are given and the basic theorems are proved, i.e., Stokes' theorem, the flow theorem, the Frobenius integration theorem, real analytic integrable almost complex implies complex, the Poincaré lemma, the Grothendieck lemma, the Oka-Weil approximation theorem, the Whitney embedding and immersion theorems, the Thom transversality theorem.

Chapter III deals with linear differential operators on vector bundles. The basic theorems on Fourier transforms (inversion formula, Plancherel, etc.) in $\mathbb{R}^{n}$ are proved.
...

The book is clearly written and would make an excellent text for a graduate course in analysis.

3.4. Review by: D Pascali.
zbMATH 0188.25803.

As the preface shows, this monography is based on the author's lectures given at the Tata Institute of Fundamental Research, Bombay in 1964/65 dealing with some topics in analysis, both on real and complex manifolds. One must firstly notice, that the work is accessible to the readers who are acquainted with linear and multilinear algebra, topology, theory of real and complex functions. In this aim, in the first of the three chapters of the book, some special properties of the differentiable functions on $R^{n}$, which are often used in differential topology, are studied. There are given complete proofs of the implicit function theorem and of other related theorems; of the Sard's theorem with applications to functional dependence; of Whiteney's approximation theorems and of the existence and uniqueness of solutions of ordinary differential equations. Analogue results for holomorphic mappings are mentioned. The second chapter, the most important and wide of the work, treats real and complex manifolds. After giving the usual definitions of vector fields, differential forms, submanifolds and integration on orientable manifolds, the author proves some basic results as Frobenius theorem, the lemmata of Poincare and Grothendieck, the Whitney's theorems on immersion and imbeddings, the Thom's transversality theorem. Complex and almost complex manifolds are also studied and the integrability theorem of an almost complex structure is proved in the case when the given almost complex structure is real analytic. The end of the book is illustrated by the theory of linear differential elliptic operators on manifolds, exposed in the third chapter. This starts with the definitions of vector bundles, the Fourier transforms and characterisations of linear differential operators due to Peetre and Hörmander. The lemmata of Rellich and Sobolev are proved as well as the inequalities of Garding and Friedrichs. They are used to prove the regularity theorem of weak solutions of elliptic equations. In the last paragraph the approximation theorem of Malgrange-Lax is proved and then is used the proof of the Runge theorem on open Riemann surfaces. The volume can be thought as a textbook whose content is carefully selected introducing the reader in an important domain of research of modern mathematics.

4.1. From the Preface.

There are at least three parts of the theory of functions of several complex variables which are of importance in various branches of m mathematics. These are:

(1) the elementary theory, comprising Hartogs' theory, domains of holomorphy, and automorphisms of bounded domains;

(2) the local and global study of analytic sets and complex spaces;

and

(3) global ideal theory, Stein manifolds, coherent analytic sheaves.

Of these the third has been dealt with in the books of Hörmander and of Gunning and Rossi and in several sets of lecture notes ..

While parts of the elementary theory are treated in most books or lecture notes I have felt that a reasonably complete account has been lacking, and I hope that the present notes will provide access to some of the most important parts of this aspect of the theory.

The material in these notes was presented in lectures during the autumn quarter of 1969 at the University of Chicago. Much of it I covered in lectures in 1967 and 1968 at the University of Geneva. The notes taken by Pierre Siegfried at Geneva have made the task of pre-paring the present notes much easier.

The only prerequisites are the following: elements of the theory of functions of one complex variable and of measure theory; point set topology; the implicit function and rank theorems; and, in the last chapter, the local existence and uniqueness theorems in the theory of ordinary differential equations.

4.2. Review by: J Kajiwara.
Mathematical Reviews MR0342725 (49 #7470).

This book, a revision and organisation of lectures given by the author at the University of Geneva in 1967 and 1968 and at the University of Chicago in 1969, is a concise, excellent and well-written introduction to the elementary theory of functions of several complex variables, which is of importance not only in various branches of mathematics but also in theoretical physics. The only prerequisites are the following: elements of the theory of functions of one complex variable and of measure theory; point set topology; the implicit function and rank theorems; and, in the last chapter, local existence and uniqueness theorems from the theory of ordinary differential equations. Readers may enjoy novel proofs of classical theorems here and there.

Chapter 1 contains elementary properties of functions of several complex variables, including the definition of holomorphic functions by power series expansions, Cauchy's formula, the open mapping theorem and Weierstrass' and Montel's theorems. The maximum principle is given as an application of the open mapping theorem. The author proves Montel's theorem not by using the Arzelà-Ascoli theorem but by using power series expansions and the diagonal method. Chapter 2 discusses analytic continuation, including Laurent and power series expansions in Reinhardt domains, the definition of germs of holomorphic functions, of the sheaf $\mathcal{O}$ and of unramified domains in $\mathbb{C}^{n}$, the monodromy theorem, the Poincaré-Volterra theorem and the principle of analytic continuation. Cauchy's theorem on the $\bar{\partial}$-problem in a simply connected domain in $\mathbb{C}^{n}$ is given as an application of the monodromy theorem. Chapter 3 includes the definitions and basic properties of harmonic and subharmonic functions, Hartogs' theorem on a separate analyticity and exceptional sets of subharmonic functions. The object of Chapter 4 is to prove a theorem due to Hartogs to the effect that the set of singularities of a holomorphic function tends to be analytic. The object of Chapter 5 is a discussion of the automorphisms of bounded domains, including Cartan's uniqueness theorem, automorphisms of circular domains, Poincaré's theorem that the polydisc and the ball are analytically distinct, proper holomorphic maps, the theorem of Remmert-Stein and some generalisations of it ...

4.3. Review by: S Hitotumatu.
zbMATH 0223.32001.

This book is due to lectures at Chicago and Geneva by the author. In the preface, he emphasises a reasonably complete amount of things lacking within the elementary theory in most books so far. He begins with elementary properties such as Cauchy's formula, open mapping theorem for a holomorphic function, normal family and analytic continuation in the first two chapters. Then he discusses subharmonic functions including Hartogs' theorem on separately holomorphic functions, and Hartogs' theorem on the singularity of holomorphic functions. In Chapter 5 titled "automorphism of bounded domain", the author defines $Aut(D)$, the topological group of all holomorphic automorphisms of a bounded domain $D$ with compact-open topology. Then he discusses classical results such as Cartan's uniqueness theorem, Poincare's result, proper holomorphic mappings and limits of automorphisms. In the succeeding three chapters, he discusses envelope of holomorphy, domain of holomorphy and convexity, and Bishop's proof of Oka's theorem. As examples, he treats Reinhardt domains and tubes, and especially he gives an interesting generalisation of the results of Herve and Thullen, saying that $D$ is a domain of holomorphy if $D/Aut(D)$ is contained in a compact set. In the final Chapter 9, he begins with a concise introduction on Lie theory, and proves a theorem of H Cartan asserting that $Aut(D)$ carries a structure of Lie group and acts real analytically on $D$ provided that $D$ is a bounded domain in $\mathbb{C}^{n}$. Regrettably, there are several systematic inconsistencies of the numbers of referred papers between in the text and in the table of references. As a whole, this is a very handy text book to the introduction of the theory of several complex variables also for undergraduate students.

5.1. From the Preface.

This book is based on a first-year graduate course I gave three times at the University of Chicago. As it was addressed to graduate students who intended to specialise in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Complex analysis in one variable is ideally suited to this attempt. Of course, the branches of mathematics one chooses, and the connections one makes, must depend on personal taste and knowledge. My own leaning towards several complex variables will be apparent, especially in the notes at the end of the different chapters.

The first three chapters deal largely with classical material which is available in the many books on the subject. I have tried to present this material as efficiently as I could, and, even here, to show the relationship with other branches of mathematics.

Chapter 4 contains a proof of Picard's theorem; the method of proof I have chosen has far-reaching generalisations in several complex variables and in differential geometry.

The next two chapters deal with the Runge approximation theorem and its many applications. The presentation here has been strongly influenced by work on several complex variables.

Although Chapter 8 is entitled "Functions of several complex variables," the book as a whole is about a single variable. This chapter is meant to contrast the behaviour in higher dimension with that in the plane.

Chapter 9, on Riemann surfaces, is meant to serve as an introduction to tools which are of great importance, not only in the modern study of Riemann surfaces, but also in algebraic geometry, in several complex variables and elsewhere.

Chapter 10 presents Tom Wolff's proof of the corona theorem. It is meant to demonstrate the use of real variable methods in complex analysis, and could be used as an introduction to the study of $H^{p}$ spaces, a subject that has been much in evidence in recent work in Fourier analysis.

The last chapter is a return to classical material. Subharmonic functions and their generalisations to several variables are of great importance.

There are notes at the end of each chapter which are partly historical and partly an attempt to point out some directions in which the material of the chapter has developed.
...

As for prerequisites, it is assumed that the reader is well acquainted with calculus in several variables and with point set topology (properties of locally compact spaces, of connected components and the like). He will need some basic definitions and results from linear algebra and the theory of rings and ideals, as he will elementary properties of Lebesgue measure and the standard convergence theorems for Lebesgue integrals. Finally, from functional analysis, he will need the Hahn-Banach theorem and the closed graph theorem (for Banach spaces) and a few other elementary and easily accessible facts (such as the finite dimensionality of a locally compact Banach space).

I prepared a handwritten version of the course I gave at Chicago. A few people who saw this version (W Beckner, K Chandrasekharan, I Kaplansky among them) found it useful and suggested that it might be of some general use. I am very grateful to them for their encouragement and for the many suggestions they made.

I am also grateful to Klas Diederich who saw my handwritten notes and suggested to Klaus Peters that Birkhäuser might be interested in publishing this book.

To K Chandrasekharan and Irving Kaplansky, I owe special thanks. Without the suggestions and encouragement of the former, this book would never have been completed. As for the latter, anyone glancing through the notes on Chapters 6 and 9 will find acknowledgment for specific results and proofs he showed me. But he also read my notes very carefully, compared the proofs given there with others in the literature and was always most helpful with suggestions, references, and answers to questions.

5.2. Review by: David Minda.
Mathematical Reviews MR0781130 (87h:30001).

This book provides an alternative for a first-year graduate course in the classical theory of functions of one complex variable. A theme of the book is to relate classical complex analysis to other branches of mathematics. It includes many of the standard topics for a basic graduate course, but the exposition and viewpoint are strongly influenced by the theory of several complex variables. In fact, there is even a brief chapter dealing with functions of several complex variables; this is used to show that their behaviour is sometimes quite different from functions of one complex variable. One pleasant feature of the text is an early and elementary treatment of the theorems of Picard, Landau and Schottky via Ahlfors' extension of Schwarz's lemma in Chapter 4. In addition to covering many of the standard topics, the author also provides a treatment of covering spaces, the inhomogeneous Cauchy-Riemann equation, compact Riemann surfaces and Wolff's proof of the corona theorem. Overall, the author's approach is analytic rather than geometric. Some classical topics not covered include Möbius transformations, elliptic functions and entire functions. The lack of exercise sets will hinder the use of the book as a text. Also, there is a substantial (for a typical first-year graduate course) list of prerequisites: multivariable calculus, point set topology, elementary Lebesgue integration and elementary functional analysis.

5.3. Review by: M von Rentein.
zbMATH 0561.30001.

This book is not a textbook on function theory in the usual sense. In many respects, it goes far beyond conventional presentations. When considering the presentation and selection of material, it is important to bear in mind that the book was written by a researcher specialising in functions of several complex variables. Accordingly, the selection is specific, and it is understandable that some classical components are not covered, such as Möbius transformations or simple functions. However, the book offers a wealth of material not found in previous textbooks on function theory. The book consists of 11 chapters. Each chapter has an appendix with remarks, mostly of a historical nature, and its own bibliography. In the first third of the book (chapters 1, 2, and 3), the book covers those parts of the theory that are also found in several, especially more recent, textbooks. However, even here there are exceptions. Thus, the existence of an antiderivative is deduced from the vanishing of the first cohomology group, covering surfaces and the monodromy theorem are developed in the context of manifolds, the sheaf of germs of holomorphic functions is introduced, and finally, the Looman-Menchoff theorem is carefully and comprehensively proven via numerous lemmas. The consistent use of partial derivatives with respect to $z$ and $\bar {z}$ (Wirtinger calculus) is a given.

In Chapter 4, Picard's grand theorem is proven using Schottky's theorem. The well-known Montel normality criterion for a family of functions that omits two values ​​in the image domain is also presented. In the appendix, the author points to the further development by R Nevanlinna and even provides references to Nevanlinna's theory in several complex variables. Chapter 5 deals with the particularly important inhomogeneous Cauchy-Riemann differential equations, which play a major role in modern developments, and concludes with Runge's approximation theorem in various versions and the homology form of Cauchy's integral theorem. The next chapter deals with the Mittag-Leffler and Weierstrass theorems and presents a cohomology form of Cauchy's integral theorem.
...

Besides a solid understanding of differential calculus of several (real) variables (which is a given), the book only requires basic knowledge of topology and measure theory. However, it is not suitable for beginners who are just starting out with complex analysis. It lacks motivation, examples, and the interpretation of theorems. The proofs are also generally very concise. On the other hand, it is all the more suitable for an advanced course. It sheds new light on some familiar concepts (multiple variables) and reveals that it is not just about complex analysis, but truly complex analysis (of one variable). For many purely complex problems (e.g., the Corona problem), it is extremely advantageous, useful, and, so far, essential to work with nonholomorphic functions in the solution process. Furthermore, the book demonstrates numerous connections to other areas of mathematics. Overall, it continues the path already taken by leading textbooks on complex analysis published in recent years.

6.1. From the Preface.

The present edition of this book is simply a reprint of the second (1973) with such misprints corrected as I have noticed. I should like to take this opportunity to make a few general comments on the contents, and mention an alternative approach to the theory of linear elliptic operators (Chapter 3).

6.2. Review by: J Lorenz.
zbMATH 0583.58001.

This is a reprint of a book, which has become almost a classic and which has been out of print for a long time. The new edition contains an additional preface with general comments on the contents, some historical remarks on "Poincaré's lemma" (Poincaré had nothing to do with this result, it should be attributed to Volterra) and remarks on an alternative approach to the theory of linear elliptic differential operators on complex vector bundles, based on pseudodifferential operators with symbol from the class $S^{m}$, all with complete references (21 items).

Although its first printing appeared almost 20 years ago (1968) this book has remained up to date and modern. It constitutes an excellent text introducing global analysis.

7.1. From the Preface.

These notes form the contents of a Nachdiplomvorlesung given at the Forschungsinstitut für Mathematik of the Eidgenössische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof K Chandrasekharan and Prof Jurgen Moser have encouraged me to write them up for inclusion in the series, published by Birkhäuser, of notes of these courses at the ETH.

Dr Albert Stadler produced detailed notes of the first part of this course, and very intelligible classroom notes of the rest. Without this work of Dr Stadler, these notes would not have been written. While I have changed some things (such as the proof of the Serre duality theorem, here done entirely in the spirit of Serre's original paper), the present notes follow Dr Stadler's fairly closely.

My original aim in giving the course was twofold. I wanted to present the basic theorems about the Jacobian from Riemann's own point of view. Given the Riemann-Roch theorem, if Riemann's methods are expressed in modern language, they differ very little (if at all) from the work of modern authors.

I had hoped to follow this with some of the extensive work relating theta functions and the geometry of algebraic curves to solutions of certain non-linear partial differential equations (in particular KdV and KP). Time did not permit pursuing this subject, and I have contented myself with a couple of references in §17. These references fail to cover much other important work (especially of M Mulase) but I have not tried to do better because the literature is so extensive.

It is a great pleasure to express my thanks to the ETH for its hospitality, to Prof J Moser for his encouragement, and to Dr A Stadler for the enormous amount of work he undertook which made these notes easier to write. But special thanks are due to Prof K Chandrasekharan. But for him, I would not have been at the ETH, nor would
these notes have been written without his advice and encouragement.

7.2. Review by: R F Lax.
Mathematical Reviews MR1172116 (93j:30049).

This book consists of lecture notes from a course given at the ETH in Zürich. In the preface, the author states that one of his aims was "to present the basic theorems about the Jacobian from Riemann's own point of view. Given the Riemann-Roch theorem, if Riemann's methods are expressed in modern language, they differ very little (if at all) from the work of modern authors."

The first four chapters of the book deal with algebraic functions, Riemann surfaces defined as connected 2-manifolds with a complex structure, germs of holomorphic functions, and the Riemann surface of an algebraic function. The next six chapters deal with sheaves, vector bundles, finiteness theorems, the Dolbeault isomorphism for the Čech cohomology groups $H^{0}$ and $H^{1}$, the Weyl lemma for the operator $\bar{\partial}$, Serre duality and the Riemann-Roch theorem. While almost all arguments are self-contained, the author assumes a solid foundation. (It would probably be helpful to a student if he or she had learned complex analysis from the author's earlier book [Complex analysis in one variable, 1985].) The pace is quite brisk - the Riemann-Roch theorem is proved on page 51. The treatment is similar to that in R C Gunning's book [Lectures on Riemann surfaces, 1966] and in O Forster's text [Riemannsche Flächen, 1977]. There are also similarities to the author's previous monograph with J Guenot, although noncompact surfaces are not considered in the work under review.

The next two chapters are concerned with the Riemann-Hurwitz formula, Weierstrass points, hyperelliptic surfaces, and canonical embeddings. In the following chapter, the author presents some geometry of curves in projective space, including the geometric form of the Riemann-Roch theorem, Castelnuovo's general position theorem, and Clifford's theorem. The next two chapters deal with Riemann's bilinear relations, the Jacobian, and Abel's theorem. The final four chapters treat the theta function, the theta divisor, Torelli's theorem, and Riemann's theorem on the singularities of the theta divisor.

All of the above topics are presented in about 115 pages. The rapid pace may make these notes tough going for someone who is seeing this material for the first time. Also, there are no exercises and no index in this book.

A student would probably benefit most by using these notes in conjunction with one of the more standard texts.

7.3. Review by: C Andreian Cazacu.
zbMATH 0758.30002.

This remarkable book is distinguished from other books treating the subject using methods of algebraic topology and several complex variables, by the contents which, besides the usual results, includes special theorems not presented in other monographs and by the choice of the proofs, as well as by the conciseness and elegance of the style.

It also presents a brilliant survey of the present interest of Riemann's ideas and methods, which "expressed in modern language, differ very little (if at all) from the work of modern authors" (preface). Thus the Jacobian and the singularities of the theta divisor are presented from Riemann's own point of view. Moreover, Serre's duality is exposed in the spirit of Serre's original paper, an interesting example of a Riemann surface is presented quoting a paper of de Rham. In the same time the book opens the way to most of the recent research topics and selective references stimulate the reader to pursue the study in these directions.

The preliminary part of the book contains: §1. Algebraic functions, proper maps, coverings; §2. Riemann surfaces; §3. The sheaf of germs of holomorphic functions on Riemann surfaces, the Riemann surfaces of an analytic - and in §4 an algebraic - function; §5. Sheaves, cohomology, Leray's and Mittag-Leffler's theorems; §6. Vector and line bundles, divisors.

A second cycles of paragraphs culminates in a Riemann-Roch theorem: §7. Finiteness theorems (the new proof avoids Schwartz's theorem), consequences for meromorphic sections in holomorphic line bundles and meromorphic functions; §8. Dolbeault's isomorphism, the canonical bundle; §9. Weyl's lemma, Serre's duality for holomorphic vector bundles; §10. The Riemann-Roch theorem and applications: vanishing theorems for cohomological groups, duality pairing, residue version of Serre's duality; §11. Riemann-Hurwitz formula, topological invariance of the genus, Weierstrass points, gap theorems; §12. Hyperelliptic surfaces. The connections to algebraic geometry constitute another quality of the book: Imbedding theorems in $P^{N}$ and the canonical map lead to the interpretation of the compact Riemann surfaces as a projective curve and §13 deals with the geometry of these curves: theorems by Bertini, Castelnuovo, Clifford, M Noether, a geometric form of the Riemann-Roch theorem.

The final part of the book is dedicated to the Jacobian of a compact Riemann surface.: §14. Riemann's bilinear relations; §15. The Jacobian, the Abel-Jacobi map, Abel's theorem; §16. Automorphy factors, Riemann's theta function, Lefschetz's embedding theorem; §17. The theta divisor $\theta$, the Jacobi inverse problem, Riemann's factorisation theorem, functions with prescribed poles or essential singularities; §18. Torelli's theorem (H Martens' proof); §19. Riemann's theorem on the singularities of $\theta$, other results on meromorphic functions and on quadrics.

Some familiarity with concepts and methods utilised in the book especially with the algebraic geometry, facilitate its study, but the reader's effort is rewarded by the deep acquaintance with this beautiful field of mathematics.

8.1. From the Publisher.

New to this second edition is a collection of over 100 pages worth of exercises, problems, and examples giving students an opportunity to consolidate their command of complex analysis and its relations to other branches of mathematics, including advanced calculus, topology, and real applications.

The main part of the text is a presentation of complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Looman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions on a domain in C is studied.

Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires a working knowledge of multivariable calculus, point set topology, elementary Lebesgue integration, and elementary functional analysis. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic function as on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions.

8.2. Preface to the Second Edition.

The original edition of this book has been out of print for some years. The appearance of the present second edition owes much to the initiative of Yves Nievergelt at Eastern Washington University, and the support of Ann Kostant, Mathematics Editor at Birkhäuser.

Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. In 1997, Yves Nievergelt informed me that, for a decade, he had regularly taught a course at Eastern Washington based on the book, and that he had systematically compiled exercises for his course. He kindly put his work at my disposal.

Thus, the present edition appears in two parts. The first is essentially just a reprint of the original edition. I have corrected the misprints of which I have become aware (including those pointed out to me by others), and have made a small number of other minor changes.

The second part of the book, authored by Yves Nievergelt, consists of exercises and relevant references. Most of the exercises are based on his course at Eastern Washington, but it also includes several problems from a set that I sent him. This set was a selection from problems that Kevin Corlette, Madhav Nori and I prepared when we taught the first year graduate course on one complex variable here at Chicago. We hope that the addition of this Part 2 will enhance the usefulness of the book.

The first edition of this book was dedicated to K Chandrasekharan. The reasons, professional and personal, for doing this have only grown stronger. I should like, therefore, to dedicate Part 1 of this Second Edition once again to him.

8.3. Review by: Steve Abbott.
The Mathematical Gazette 86 (505) (2002), 190.

This book offer a fresh look at complex analysis for graduate students with a basic knowledge of multi-variable calculus, linear algebra, ring theory, point-set topology, Lebesgue integration and functional analysis. Narasimhan has research interests in the theory of several complex variables, and adopts an approach that supports the transition from single variable theory.

The first part of the book is a corrected version of Narasimhan's first edition. An introduction to holomorphic functions precedes chapters on the monodromy and residue theorems, Runge's theorem and its applications, and the Riemann mapping theorem. A chapter contrasting the behaviour of functions of one and several complex variables is then followed by consideration of Riemann surfaces, the corona theorem, subharmonic functions and the Dirichlet problem. What makes the current book worthwhile even for owners of the first edition is the new second part, written by Nievergelt. This consists of hundreds of exercises, organised under the same headings and sub-headings as the first part, and supported by relevant definitions and references. A student who masters this book will have a secure foundation for further graduate work in complex analysis.

8.4. Review by: P Lappan.
Mathematical Reviews MR1803086 (2002e:30001).

This very interesting textbook on complex analysis has an unusual organisation. It is divided into two parts, where the first part of the book is the basic text written by the first author, and the second part consists of exercises by the second author, where, for example, a chapter in Part 2 consists of problems based on the material in the same numbered chapter in Part 1. (There is a preliminary Chapter 0 in Part 2 where there are exercises on material that could be considered prerequisite to the text.) Part 2 takes up about half as many pages as Part 1.

The order in which material is presented differs from that of many other texts on the subject. For example, the big Picard theorem is presented relatively early (Chapter 4), while the Riemann mapping theorem comes much later (Chapter 7). Harmonic functions are not dealt with in any meaningful way until the last chapter.
...
The exercises in Part 2 vary from basic to advanced, and provide good practice for the concepts and techniques of the subject. Some exercises take up topics that are not part of the text material in Part 1, for example, stereographic projection, Julia sets, and doubly periodic functions. There is little mention of elementary conformal mapping in the text, although there is some treatment of fractional linear transformations in the preliminary exercise chapter.

The choice of topics covered gives an excellent introduction to modern complex analysis. The exposition is well written. All in all, this book is a welcome addition to the list of books presenting a first course in complex analysis.

The chapter headings (which are the same in both parts) are: Chapter 1. Elementary theory of holomorphic functions; Chapter 2. Covering spaces and the monodromy theorem; Chapter 3. The winding number and the residue theorem; Chapter 4. Picard's theorem; Chapter 5. Inhomogeneous Cauchy-Riemann equation and Runge's theorem; Chapter 6. Applications of Runge's theorem; Chapter 7. Riemann mapping theorem and simple connectedness in the plane; Chapter 8. Functions of several complex variables; Chapter 9. Compact Riemann surfaces; Chapter 10. The corona theorem; Chapter 11. Subharmonic functions and the Dirichlet problem. In addition, Part 2 also has a Chapter 0 - Review of complex numbers. At the end of each chapter in the first part there are ``notes'' that point out some of the history of the subject, alternative approaches to some of the material (with references), and specific attributions for suggestions the first author received from others.

8.5. Review by: Joachim Naumann.
zbMATH 1009.30001.

The first part of the book under review represents essentially the material of R. Narasimhan's Complex analysis in one variable (first edition, 1985). The second part of the book, authored by Y. Nievergelt, consists of exercises and relevant references. As for the prerequisites, the reader is assumed to be well acquainted with the calculus of several variables and with point set topology. Moreover, some basic facts from linear algebra and the theory of rings and ideals are needed, as well as elementary properties of Lebesgue measure and integral, and basic results from functional analysis.

The table of contents is as follows.

I. Complex analysis of one variable. 1. Elementary theory of holomorphic functions. 2. Convering spaces and the monodromy theorem. 3. The winding number and the residue theorem. 4. Picard's theorem. 5. Inhomogeneous Cauchy-Riemann equation and Runge's theorem. 6. Applications of Runge's theorem. 7. Riemann mapping theorem and simple connectedness in the plane. 8. Functions of several complex variables. 9. Compact Riemann surfaces. 10. The corona theorem. 11. Subharmonic functions and the Dirichlet problem.

II. Exercises. The second part of the book presents more than 300 worked exercises the level of which varies from straight-forward to challenging. The exercises are arranged in complete coincidence with the material of the chapters of part I. Some of the exercises point to theorems which supplement the theory presented in part I.

The first three chapters of part I deal largely with classical topics which are available in many books on functions of a complex variable. Chapter 4 contains a proof of Picard's theorem; the method of proof has important generalisations in several complex variables and in differential geometry. The next two chapters are concerned with the Runge approximation theorem and some of its many applications. Chapters 7 and 9 contain standard material. Chapter 8 demonstrates some contrasts between functions of one complex variable and functions of several complex variables. Chapter 10 presents T Wolff's poof of the corona theorem. The discussion in this chapter illustrates the use of real variable methods in complex analysis, and could be used as an introduction to the study of $H^{p}$ spaces. The last chapter is a return to classical material.

There are notes at the end of each chapter which contain brief remarks on the history of the material presented as well as references to the literature.

The exercises of part II give the reader the opportunity to consolidate his knowledge in complex analysis. At the end of this part there are notes for the exercises and references.

The book can be highly recommended for a thorough study of complex analysis.