1st Edition
An Introduction to Complex Analysis Classical and Modern Approaches
Like real analysis, complex analysis has generated methods indispensable to mathematics and its applications. Exploring the interactions between these two branches, this book uses the results of real analysis to lay the foundations of complex analysis and presents a unified structure of mathematical analysis as a whole.
To set the groundwork and mitigate the difficulties newcomers often experience, An Introduction to Complex Analysis begins with a complete review of concepts and methods from real analysis, such as metric spaces and the Green-Gauss Integral Formula. The approach leads to brief, clear proofs of basic statements - a distinct advantage for those mainly interested in applications. Alternate approaches, such as Fichera's proof of the Goursat Theorem and Estermann's proof of the Cauchy's Integral Theorem, are also presented for comparison.
Discussions include holomorphic functions, the Weierstrass Convergence Theorem, analytic continuation, isolated singularities, homotopy, Residue theory, conformal mappings, special functions and boundary value problems. More than 200 examples and 150 exercises illustrate the subject matter and make this book an ideal text for university courses on complex analysis, while the comprehensive compilation of theories and succinct proofs make this an excellent volume for reference.
The field of complex numbers
The complex plane
Metric spaces
Mappings and functions. Continuity
THE CLASSICAL APPROACH
Ordinary complex differentiation
Preliminaries of the Integral Calculus
Complex Integral Theorems
AN ALTERNATIVE APPROACH
Partial complex differentiations
Complex Green-Gauss Integral Theorems
Generalized Cauchy Integral Formula
The classical Cauchy Integral Formula
Comparison
LOCAL PROPERTIES
Existence of higher order derivatives
Local power series representation
Distribution of zeros
The Weierstrass Convergence Theorem
Connection with plane Potential Theory
Complex Integral Theorems revisited
GLOBAL PROPERTIES
Analytic continuation
Maximum Modulus Principle
Entire functions
Fundamental Theorem of Algebra
ISOLATED SINGULARITIES
Classification
Laurent series
Characterization by the principal part
Meromorphic functions
Behavior at essential singularities
Behavior at infinity
Partial fractions of rational functions
Meromorphic functions on the Sphere
HOMOTOPY
Statement of the problem
Homotopic curves
Path independent line integrals
Simply connected domains
Solution of first order systems
Conjugate solutions
Inversion of complex differentiation
Morera's Theorem
Potentials of vector fields
RESIDUE THEORY
Statement of the problem
Winding numbers
The integration of principal parts
Residue Theorem
Calculation of residues
APPLICATIONS OF RESIDUE CALCULUS
Total number of zeros and poles
Evaluation of definite integrals
Sum of certain series
MAPPING PROPERTIES
Continuously differentiable mappings
Conformal mappings
Examples of conformal mappings
Univalent functions
Riemann's Mapping Theorem
Construction of flow lines
SPECIAL FUNCTIONS
Prescribed principal parts
Prescribed zeros
Infinite products
Weierstrass products
Gamma Function
The Riemann Zeta Function
Elliptic Functions
BOUNDARY VALUE PROBLEMS
Preliminaries
The Poisson Integral Formula
Cauchy Type Integrals
Desired Holomorphic Functions
Biography
Wolfgang Tutschke, Harkrishan L. Vasudeva
"Many things, which are briefly described in others books, in remarks or exercises, are given in full detail … . [It] will please readers interested … in applications as well as those who want to know how things really work and prefer deeper and more detailed treatment of the material. The book also contains more than 200 examples and 150 exercises. … I recommend it for courses in complex function theory … and also as a reference book."
- EMS Newsletter, Dec. 2004
"… [A]bundant examples and 'hints' to aid readers [are provided]. Summing Up: Recommended. Upper-division undergraduates through professionals."
- CHOICE, March 2005, Vol. 42, No. 07
"For the unification of the structure of mathematical analysis as a whole, it is imperative to use results of real analysis when laying the foundations of complex analysis. This is done in the present book."
-Zentralblatt MATH
