The idea for this book came when I was an assistant at the Department of Mathematics and Computer Science at the Philipps-University Marburg, Germany. Several times I faced the task of supporting lectures and seminars on complex analysis of several variables and found out that there are very few books on the subject,compared to the vast amount of literature on function theory of one variable, let alone on real variables or basic algebra. Even fewer books, to my understanding,were written primarily with the student in mind. So it was quite hard to find supporting examples and exercises that helped the student to become familiar with the fascinating theory of several complex variables.
preface
1 elementary theory of several complex variables
1.1 geometry of cn
1.2 holomorphic functions in several complex variables
1.2.1 definition of a holomorphic function
1.2.2 basic properties of holomorphic functions
1.2.3 partially holomorphic functions and the cauchy-pdemann differential equations
1.3 the cauchy integral formula
1.4 o (u) as a topological space
1.4.1 locally convex spaces
1.4.2 the compact-open topology on c (u, e)
1.4.3 the theorems of arzela-ascoli and montel
1.5 power series and taylor series
1.5.1 summable families in banach spaces
1.5.2 power series preface
1 elementary theory of several complex variables
1.1 geometry of cn
1.2 holomorphic functions in several complex variables
1.2.1 definition of a holomorphic function
1.2.2 basic properties of holomorphic functions
1.2.3 partially holomorphic functions and the cauchy-pdemann differential equations
1.3 the cauchy integral formula
1.4 o (u) as a topological space
1.4.1 locally convex spaces
1.4.2 the compact-open topology on c (u, e)
1.4.3 the theorems of arzela-ascoli and montel
1.5 power series and taylor series
1.5.1 summable families in banach spaces
1.5.2 power series
1.5.3 reinhardt domains and laurent expansion
2 continuation on circular and polycircular domains
2.1 holomorphic continuation
2.2 representation-theoretic interpretation of the laurent series
2.3 hartogs' kugelsatz, special case
3 biholomorphic maps
3.1 the inverse function theorem and implicit functions
3.2 the riemann mapping problem
3.3 cartan's uniqueness theorem
4 analytic sets
4.1 elementary properties of analytic sets
4.2 the riemann removable singularity theorems
5 hartogs' kugelsatz
5.1 holomorphic differential forms
5.1.1 multilinear forms
5.1.2 complex differential forms
5.2 the inhomogenous cauchy-riemann differential equations
5.3 dolbeaut's lemma
5.4 the kugelsatz of hartogs
6 continuation on tubular domains
6.1 convex hulls
6.2 holomorphically convex hulls
6.3 bochner's theorem
7 cartan-thullen theory
7.1 holomorphically convex sets
7.2 domains of holomorphy
7.3 the theorem of cartan-thullen
7.4 holomorphically convex reinhardt domains
8 local properties of holomorphic functions
8.1 local representation of a holomorphic function
8.1.1 germ of a holomorphic function
8.1.2 the algebras of formal and of convergent power series
8.2 the weierstrass theorems
8.2.1 the weierstrass division formula
8.2.2 the weierstrass preparation theorem
8.3 algebraic properties of c {z1,., zn}
8.4 hilbert's nullstellensatz
8.4.1 germs of a set
8.4.2 the radical of an ideal
8.4.3 hilbert's nullstellensatz for principal ideals register of symbols
bibliography
index
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