VIII – Cauchy Theory
§ 1. Integrals of holomorphic functions – § 2. Cauchy’s Integral
Formulas – § 3. Some Applications of Cauchy’s Method
In Chapter VII, § 4 we showed how a significant part of the classical theory
of holomorphic or analytic functions on C can be obtained from Fourier series.
In fact, our universal method for constructing them – a fundamental idea
of Cauchy – is to integrate holomorphic functions along curves drawn in
their domains of definition and thereby obtain a version of the “ fundamental
theorem of differential and integral calculus ” (FT) for holomorphic functions,
and then to deduce countless consequences.
I will present only very few of them. The general theory of analytic functions is of unlimited scope
1 and the results needed in mathematical fields
where holomorphic functions are encountered are, on the other hand, very
limited in most cases. For instance, a famous result like Riemann’s theorem
on conformal mapping of simply connected domains is rarely used, although it
is recommended to be familiar with it for the sake of “ general knowledge ”;
as for classifying simply connected Riemann surfaces, which would be far
more useful, it would need far too complicated developments. The very basic
results and methods that we will present in this chapter are, for example,
quite sufficient for the chapter devoted to the theory of Riemann surfaces or
for the one on elliptic and modular functions.
1 The two volumes by Reinhold Remmert, Funktionentheorie (Springer, 1995, also
available in English edition), more than 700 very compact pages, can give some
idea of the general theory of analytic functions, but do not cover Riemann surfaces, elliptic and automorphic functions, differential equations in the complex
domain, special functions, etc., areas that would require thousands of additional
pages and that, at any rate, have been the subject of specialized presentations.
Other numerous available presentations include Walter Rudin, Real and Complex
Analysis (McGraw-Hill, 1966, also available in French), Jean Dieudonn´ e, Calcul
Infinit´ esimal (Hermann, 1968), in particular useful for its many exercises, Eberhard Freitag & Rolf Busam, Funktionentheorie (Springer-Verlag, 1995), which
lists several other books, Serge Lang, Complex Analysis (Springer, many editions), John B. Conway, Functions of One Complex Variable (2 vol., Springer,
1978–95), Carlos A. Berenstein & Roger Gray, Complex Variables. An Introduction (Springer, 1991).
© Springer International Publishing Switzerland 2015
1
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_1
§ 1. Integrals of holomorphic functions – § 2. Cauchy’s Integral
Formulas – § 3. Some Applications of Cauchy’s Method
In Chapter VII, § 4 we showed how a significant part of the classical theory
of holomorphic or analytic functions on C can be obtained from Fourier series.
In fact, our universal method for constructing them – a fundamental idea
of Cauchy – is to integrate holomorphic functions along curves drawn in
their domains of definition and thereby obtain a version of the “ fundamental
theorem of differential and integral calculus ” (FT) for holomorphic functions,
and then to deduce countless consequences.
I will present only very few of them. The general theory of analytic functions is of unlimited scope
1 and the results needed in mathematical fields
where holomorphic functions are encountered are, on the other hand, very
limited in most cases. For instance, a famous result like Riemann’s theorem
on conformal mapping of simply connected domains is rarely used, although it
is recommended to be familiar with it for the sake of “ general knowledge ”;
as for classifying simply connected Riemann surfaces, which would be far
more useful, it would need far too complicated developments. The very basic
results and methods that we will present in this chapter are, for example,
quite sufficient for the chapter devoted to the theory of Riemann surfaces or
for the one on elliptic and modular functions.
1 The two volumes by Reinhold Remmert, Funktionentheorie (Springer, 1995, also
available in English edition), more than 700 very compact pages, can give some
idea of the general theory of analytic functions, but do not cover Riemann surfaces, elliptic and automorphic functions, differential equations in the complex
domain, special functions, etc., areas that would require thousands of additional
pages and that, at any rate, have been the subject of specialized presentations.
Other numerous available presentations include Walter Rudin, Real and Complex
Analysis (McGraw-Hill, 1966, also available in French), Jean Dieudonn´ e, Calcul
Infinit´ esimal (Hermann, 1968), in particular useful for its many exercises, Eberhard Freitag & Rolf Busam, Funktionentheorie (Springer-Verlag, 1995), which
lists several other books, Serge Lang, Complex Analysis (Springer, many editions), John B. Conway, Functions of One Complex Variable (2 vol., Springer,
1978–95), Carlos A. Berenstein & Roger Gray, Complex Variables. An Introduction (Springer, 1991).
© Springer International Publishing Switzerland 2015
1
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_1