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Complex Variables: An Introduction: 125
Textbooks even excellent ones are a reflection of their times Form and content of books depend on what the students know already what they are expected to learn how the subject matter is regarded in relation to other divisions of mathematics and even how fashionable the subject matter is It is thus not surprising that we no longer use such masterpieces as Hurwitz and Courant's Funktionentheorie or Jordan's Cours d'Analyse in our courses The last two decades have seen a significant change in the techniques used in the theory of functions of one complex variable The important role played by the inhomogeneous Cauchy-Riemann equation in the current research has led to the reunification at least in their spirit of complex analysis in one and in several variables We say reunification since we
think that Weierstrass Poincare and others (in contrast to many of our students) did not consider them to be entirely separate subjects Indeed not only complex analysis in several variables but also number theory harmonic analysis and other branches of mathematics both pure and applied have required a reconsidera- tion of analytic continuation ordinary differential equations in the complex domain asymptotic analysis iteration of holomorphic functions and many other subjects from the classic theory of functions of one complex variable This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written
think that Weierstrass Poincare and others (in contrast to many of our students) did not consider them to be entirely separate subjects Indeed not only complex analysis in several variables but also number theory harmonic analysis and other branches of mathematics both pure and applied have required a reconsidera- tion of analytic continuation ordinary differential equations in the complex domain asymptotic analysis iteration of holomorphic functions and many other subjects from the classic theory of functions of one complex variable This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written
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