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A companion volume to the text "Complex Variables: An Introduction"
by the same authors, this book further develops the theory,
continuing to emphasize the role that the Cauchy-Riemann equation
plays in modern complex analysis. Topics considered include:
Boundary values of holomorphic functions in the sense of
distributions; interpolation problems and ideal theory in algebras
of entire functions with growth conditions; exponential
polynomials; the G transform and the unifying role it plays in
complex analysis and transcendental number theory; summation
methods; and the theorem of L. Schwarz concerning the solutions of
a homogeneous convolution equation on the real line and its
applications in harmonic function theory.
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."
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."
The objective of this monograph is to present a coherent picture of
the almost mysterious role that analytic methods and, in
particular, multidimensional residue have recently played in
obtaining effective estimates for problems in commutative algebra.
Bezout identities, i. e., f1g1 + ... + fmgm = 1, appear naturally
in many problems, for example in commutative algebra in the
Nullstellensatz, and in signal processing in the deconvolution
problem. One way to solve them is by using explicit interpolation
formulas in Cn, and these depend on the theory of multidimensional
residues. The authors present this theory in detail, in a form
developed by them, and illustrate its applications to the effective
Nullstellensatz and to the Fundamental Principle for convolution
equations.
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