The contents of this
book include: Complex numbers, Polynomials and rational functions, Riemann
surfaces and holomorphic maps, Fractional linear transformations, Power series,
More Series, Exponential and trigonometric functions, Arcs, curves, etc, Inverse
functions and their derivatives, Line integrals, Cauchy’s theorem, The winding
number and Cauchy’s integral formula, Higher derivatives, including Liouville’s
theorem, Removable singularities, Taylor’s theorem, zeros and poles, Analysis of
isolated singularities, Local mapping properties, Maximum principle, Schwarz
lemma, and conformal mappings, Weierstrass’ theorem and Taylor series, Plane
topology, The general form of Cauchy’s theorem, Residues, Schwarz reflection
principle, Normal families, Arzela-Ascoli, Riemann mapping theorem, Analytic
continuation, Universal covers and the little Picard theorem.
This note
covers the following topics: Compactness and Convergence, Sine Function, Mittag Leffler Theorem,
Spherical Representation and Uniform Convergence.
This book explains the following topics: Introduction to Complex Number
System, Sequences of Complex Numbers, Series of Complex Number,
Differentiability, Complex Logarithm, Analytic Functions, Complex Integration,
Cauchy Theorem, Theorems in Complex Analysis, Maximum and Minimum Modulus
principle, Singularities, Residue Calculus and Meromorphic Functions, Mobius
Transformation.
Author(s): Institute of Distance and Open Learning, University of
Mumbai
In this note
the student will learn that all the basic functions that arise in calculus,
first derived as functions of a real variable, such as powers and fractional
powers, exponentials and logs, trigonometric functions and their inverses, and
also many new functions that the student will meet, are naturally defined for
complex arguments.
This note covers
the following topics: The fundamental theorem of algebra, Analyticity, Power
series, Contour integrals , Cauchy’s theorem, Consequences of Cauchy’s
theorem, Zeros, poles, and the residue theorem, Meromorphic functions and
the Riemann sphere, The argument principle, Applications of Rouche’s
theorem, Simply-connected regions and Cauchy’s theorem, The logarithm
function, The Euler gamma function, The Riemann zeta function, The prime
number theorem and Introduction to asymptotic analysis.
This note covers the following topics:
Holomorphic functions, Contour integrals and primitives, The theorems of Cauchy,
Applications of Cauchy’s integral formula, Argument. Logarithm, Powers, Zeros
and isolated singularities, The calculus of residues, The maximum modulus
principle, Mobius transformations.
This text will illustrate and
teach all facets of the subject in a lively manner that will speak to the needs
of modern students. It will give them a powerful toolkit for future work in the
mathematical sciences, and will also point to new directions for additional
learning. Topics covered includes: The Relationship of Holomorphic and Harmonic
Functions, The Cauchy Theory, Applications of the Cauchy Theory, Isolated
Singularities and Laurent Series, The Argument Principle, The Geometric Theory
of Holomorphic Functions, Applications That Depend on Conformal Mapping,
Transform Theory.
The note deals with the Basic ideas of
functions of one complex variable. Topics covered includes: Number system ,
Algebra of Complex Numbers, Inequalities and complex exponents, Functions of a
Complex Variable, Sequences and Series, Complex Integration, Consequences of
complex integration, Residue calculus, Conformal Mapping, Mapping of Elementary
transformation, Applications of conformal mapping, Further theory of analytic
functions.
Author(s): Dr.
A. Swaminathan and Dr. V. K. Katiyar
This book is designed for
students who, having acquired a good working knowledge of the calculus, desire
to become acquainted with the theory of functions of a complex variable, and
with the principal applications of that theory.Numerous examples have been given
throughout the book, and there is also a set of Miscellaneous Examples, arranged
to correspond with the order of the text.
This note covers the
following topics: basic theorems of complex analysis, infinite series, winding
numbers of closed paths in the complex plane, path integrals in the complex
plane, Holomorphic functions, Cauchys theorem, basic properties of Holomorphic
functions, applications of Cauchy's residue theorem, Elliptic functions.
This book covers the following
topics: Field of Complex Numbers, Analytic Functions, The Complex
Exponential, The Cauchy-Riemann Theorem, Cauchy’s Integral Formula, Power
Series, Laurent’s Series and Isolated Singularities, Laplace Transforms, Prime
Number Theorem, Convolution, Operational Calculus and Generalized Functions.
This note covers the following topics: Complex Numbers, Functions of
Complex Variables, Analytic Functions, Integrals, Series, Theory of Residues and
Its Applications.
This note covers the following topics: Examples of Complex Functions,
C- Differentiable Functions, Integration, Taylor Series, Laurent Series and
Residues.