an Introduction to Complex Analysis

This textbook introduces the subject of complex analysis to advanced undergraduate and graduate students in a clear and concise manner.

From the reviews of the first printing, published as Volume 7 of the Encyclopaedia of Mathematical Sciences:
"…… In this volume, we find an introductory essay entitled "Remarkable Facts of Complex Analysis" by Vitushkin… This is followed by articles by G.M.Khenkin on integral formulas in complex analysis, by E.M.Chirka on complex analytic sets, by Vitushkin on the geometry of hypersurfaces and by P.Dolbeault, on the theory of residues in several variables. … In sum, the volume under review is the first quarter of an important work that surveys an active branch of modern mathematics. Some of the individual articles are reminiscent in style of the early volumes of the first Ergebnisse series and will probably prove to be equally useful as a reference; all contain substantial lists of references."
Bulletin of the American Mathematical Society, 1991 "… This remarkable book has a helpfully informal style, abundant motivation, outlined proofs followed by precise references, and an extensive bibliography; it will be an invaluable reference and a companion to modern courses on several complex variables."

The idea for this book came when I was an assistant at the Department of Mat- matics and Computer Science at the Philipps-University Marburg, Germany. S- eraltimesIfacedthetaskofsupportinglecturesandseminarsoncomplexanalysis 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 ?nd s- porting examples and exercises that helped the student to become familiar with the fascinating theory of several complex variables. Of course, there are notable exceptions, like the books of R.M. Range [9] or B. and L. Kaup [6], however, even those excellent books have a drawback: they are quite thick and thus quite expensive for a student’s budget. So an additional motivation to write this book was to give a comprehensive introduction to the theory of several complex variables, illustrate it with as many examples as I could ?nd and help the student to get deeper insight by giving lots of exercises, reaching from almost trivial to rather challenging. There are not many illustrations in this book, in fact, there is exactly one, becauseinthetheoryofseveralcomplexvariablesI?ndmostofthemeithertrivial or misleading. The readers are of course free to have a di?erent opinion on these matters.