Functions of a Complex Variable Krishna Series PDF is available here which can be download by anyone here for free. The functions of a complex variable (krishna series) pdf helps to improve their concept of mathematics for various competition exam such as IITJAM, CSIR-NET etc as well as for their postgraduates courses.

## Overview of krishna series complex variable pdf:

Function of a complex variable (krishna series) pdf is especially designed for honours and post graduates students of various universities. The author of the book pdf is J.N Sharma who is a Ex senior lecturer of department of mathematics, Meerut college. The book is very helpful in preparation of various competition of undergraduate and post-graduate level.

The theory of functions of a complex variable pdf is written in such a easy way that can be easily understood by a beginner as well as by an expert. The theory is properly explained by using logical arguments which help you to understand the concept and make your foundation strong in mathematics.

The krishna series complex variable pdf comes with various solved and unsolved questions to practice helpful for competition exams such as IITJAM, CSIR-NET, GATE etc.

## Content of Functions of Complex Variable by Krishna Series pdf:

The book contains the following chapters and topics as given in the table below.

• CHAPTER 0: Elements of Set Theory
• Sets and Their Basic Operations.
• Relations.
• Functions.
• R as an Ordered Field.
• Convergence of Sequences in R.
• CHAPTER 1: Complex Numbers and their Geometrical Representation
• Introduction.
• Definitions.
• Fundamental Laws of Addition and Multiplication.
• Difference of Two Complex Numbers.
• Modulus and Argument of Complex Numbers.
• The Geometrical Representation of Complex Numbers.
• Vector Representation of Complex Numbers.
• The Point on the Argand Plane Representing the Sum, Difference, Product and Division of Complex Numbers.
• Conjugate Complex Numbers.
• Properties of Moduli.
• Properties of the Arguments.
• Solved Examples.
• Riemann Sphere and the Point at Infinity.
• CHAPTER 2: Analytical Functions
• Complex Differentiation.
• Limit and Continuity.
• Differentiability.
• The Necessary and Sufficient Conditions for /(z) to be Analytic.
• Method of Constructing a Regular Function.
• A Simple Method of Constructing an Analytic Function.
• Polar form of Gauchy-Riemann Equations.
• Complex Equation of a Straight Land Circle.
• Polynomials.
• Rational Functions.
• Multiple Valued Functions.
• CHAPTER 3: Power Series and Elementary Functions.
• Sequences.
• Series.
• Sequences and Series of Functions.
• Power Series.
• Solved Examples.
• Elementary Transcendental Functions.
• CHAPTER 4: Conformal Mappings
• Mapping or Transformations.
• Jacobian of a Transformation.
• Conformal Mapping.
• Sufficient Conditions for w = /(z) to Represent Conformal Mapping.
• Necessary Conditions for w = /(z) to Represent Conformal Mapping.
• Superficial Magnification.
• The Circle.
• Inverse Points with Respect to a Circle.
• Some Elementary Transformations.
• Linear Transformations.
• Bilinear Transformation.
• Resultant of two Bilinear Transformations.
• Bilinear Transformation as the Resultant of Elementary Bilinear Transformation with Simple Geometric Properties.
• Bilinear Transformations as the Resultant of an Even Number of Inversions.
• The Linear Group.
• Equation of a Circle Through Three Given Points.
• Cross-ratio.
• Preservance of Cross-ratio under Bilinear Transformations.
• To Find Bilinear Transformations which Transform Three Distinct Points into Three Specified Points.
• Preservance of the Family of Circles and Straight Lines under Bilinear Transformations.
• Two Important Families of Circles.
• Fixed Points of a Bilinear Transformation.
• Normal Form of a Bilinear Transforrnation.
• Elliptic, Hyperbolic and Parabolic Transformations.
• Some Special Bilinear Transformations.
• CHAPTER-5: More about Conformed Mappings.
• Introduction.
• The Transformation w=z
• The Transformation w=z^2
• The Inverse Transformation
• The Exponential Transformation
• The Logarithmic Transformation
• The Trigonometrical Transformation
• Some General Techniques of Conformai Mapping
• Illustrative Examples.
• CHAPTER 6: Complex Integration
• Introduction
• Some Basic Definition
• Complex Line Integrals.
• Reduction of Complex Integrals to Real Initeigiais
• Some Properties of Complex Integrals.
• An Estimation of a Complex Integral
• Line integrals as function of Arcs
• Cauchy’s Fundamental theorem
• Second Proof of Cauchy-Goursat theorem.
• A Third Proof of.Cauchy-Goursat Theorem
• Cauchy’s Integral Formula.
• Poisson’s Integral Formula of a Circle.
• Derivative of an Analytic Function.
• Higher Order Derivatives.
• Morera’s Theorem.
• Indefinite Integrals or Primitives.
• Cauchy’s Inequality.
• Liouville’s Theorem
• Expansion of Analytic functions as Power Series: Taylor and Laurent’s Theorems.
• The Zeros of an Analytic function.
• Different Types of Singularities.
• Some Theorems on Poles and OtherSingularities.
• Solved Examples.
• The Point at Infinity.
• Characterization of Rotational Functions.
• Maximum Modulus Principle.
• The Excess of Number of Zeros over number of poles of Meromorphic function
• Rouche’s Theorem.
• Schwarz Lemma.
• Inverse Function theorem.
• Fundamental Theorem of algebra
• Analytic Continuation.
• Power Series Method of Analytic Continuation.
• Schwarz’s Reflectioh Principle.
• CHAPTER 7: Calculus of Residues.
• Residue at Simple Pole.
• Residue at a Pole of Order Greater than Unity.
• Residue at Infinity.
• Cauchy’s Residue Theorem.
• Evaluation of Definite Integrals.
• Integration round of unit circle.
• Evaluation of the Integral
• Jordan’s Inequality,
• Jordan’s Lemma.
• Evaluation of the Integrals of the different form.
• Case of the Poles on the Real Axis.
• Integrals of many Valued Functions.
• Rectangular and Other Contours.
• Expansion of Meromorhic Functions
• CHAPTER 8: Uniform Convergence and Infinite Products .
• Uniform Convergence of a Sequence.
• General Principle of Uniform Convergence.
• Uniform Convergence of a Series.
• Weierstrass’s M-test.
• Hardy’s Test.
• Continuity of the Sum Function.
• Term by Term Integration.
• Analyticity of the Sum Function of a Series, Term by Term Differentiation.
• Hurwitz Theorem.
• Uniform Convergence of Power Series.
• A note on Absolute and Uniform Convergence.
• Infinite Products.
• Three Important Theorems on Infinite Products.
• The Absolute Convergence of Infinite Products.
• Uniform Convergence of Infinite Products.
• CHAPTER 9: Entire Functions
• Introduction.
• Mittag Leffler’s Theorem.
• The Weierstrass Factorization Theorem.
• Canonical Products.
• The Jenson and Poisson-Jenson Formulas.
• Growth, Order and Convergence Exponents of Entire Functions.
• The Gamma Function.

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