WJWJ Maths
Theory of Functions of Complex Variables
  • The Complex Number System and its Properties
  • The Complex Plane
  • The Triangle Inequality
  • Polar Coordinates
  • Powers and Roots
  • The Exponential Representation
  • Lines
  • Practice Problems
  • Elementary Theory of Power series
  • Sequences
  • Complex Series
  • Uniform Convergence
  • Power Series
  • Practice Problems
  • Elementary Point Set Topology of \(\mathbb {C}\)
  • Generating Open Sets
  • Closed sets
  • Compactness
  • Connectedness
  • Sequences
  • Practice Problems
  • Complex Functions
  • Functions
  • Limits of Complex Functions
  • Continuity of Complex Functions
  • Differentiation of a Complex Function
  • Analytic Functions
  • Laplace’s Equations
  • Cauchy - Riemann Equations
  • Harmonic Functions
  • Elementary Functions
  • The Logarithm Function
  • Trigonometric and Hyperbolic Functions
  • Practice Problems
  • Analytic Functions as Mappings
  • Mappings
  • Linear Transformation
  • The Identity Mapping
  • The Mapping \(w = z^c\)
  • The Mapping \(w = z^{1/2}\)
  • The Inversion Mapping \(w = \dfrac {1}{z}\)
  • The Point at Infinity
  • Fixed Points
  • Paths and Smooth Paths
  • Linear Fractional Transformations
  • Practice Problems
  • Complex Integration
  • Arc, Curve, Contour
  • Contour Integration
  • The Cauchy - Goursat Theorem
  • Antiderivatives
  • Definite Integral
  • The Cauchy Integral Formula
  • Counting Zeroes
  • Practice Problems
  • The Maximum Modulus Theorem
  • The Maximum Modulus Principle
  • Morera’s Theorem
  • Liouville’s Theorem
  • The Fundamental Theorem of Algebra
  • Cauchy Inequality
  • Schwarz Lemma
  • Practice Problems
  • Improper Integrals Using Contour Integration
  • Jordan Inequality
  • Practice Problems
  • Calculus of Residues
  • Series
  • Singularities and Zeroes
  • Residues and the Residue Theorem
  • The Argument Principle
  • Practice Problems
  • Analytic Continuation
  • Uniqueness of the Continuation
  • Continuation Along a Path
  • The Logarithm, and Why a Function Can Fail to be Single-Valued
  • Homotopy and the Monodromy Theorem
  • The Schwarz Reflection Principle
  • When Continuation is Impossible: Natural Boundaries
  • Practice Problems
  • Problems Beyond These Notes
  • The Graduate Syllabus Behind These Problems
  • References

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4 Complex Functions

4.1 Functions
4.2 Limits of Complex Functions
4.3 Continuity of Complex Functions
4.4 Differentiation of a Complex Function
4.5 Analytic Functions
4.6 Laplace’s Equations
4.7 Cauchy - Riemann Equations
4.8 Harmonic Functions
4.9 Elementary Functions
4.10 The Logarithm Function
4.11 Trigonometric and Hyperbolic Functions
4.12 Practice Problems

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