A Guide to Complex Variables -- Preface -- Contents -- 1 The Complex Plane -- 1.1 Complex Arithmetic -- 1.1.1 The Real Numbers -- 1.1.2 The Complex Numbers -- 1.1.3 Complex Conjugate -- 1.1.4 Modulus of a Complex Number -- 1.1.5 The Topology of the Complex Plane -- 1.1.6 The Complex Numbers as a Field -- 1.1.7 The Fundamental Theorem of Algebra -- 1.2 The Exponential and Applications -- 1.2.1 The Exponential Function -- 1.2.2 The Exponential Using Power Series -- 1.2.3 Laws of Exponentiation -- 1.2.4 Polar Form of a Complex Number -- 1.2.5 Roots of Complex Numbers -- 1.2.6 The Argument of a Complex Number -- 1.2.7 Fundamental Inequalities -- 1.3 Holomorphic Functions -- 1.3.1 Continuously Differentiable and Ck Functions -- 1.3.2 The Cauchy-Riemann Equations -- 1.3.3 Derivatives -- 1.3.4 Definition of Holomorphic Function -- 1.3.5 The Complex Derivative -- 1.3.6 Alternative Terminology for Holomorphic Functions -- 1.4 Holomorphic and Harmonic Functions -- 1.4.1 Harmonic Functions -- 1.4.2 How They are Related -- 2 Complex Line Integrals -- 2.1 Real and Complex Line Integrals -- 2.1.1 Curves -- 2.1.2 Closed Curves -- 2.1.3 Differentiable and C^k Curves -- 2.1.4 Integrals on Curves -- 2.1.5 The Fundamental Theorem of Calculus along Curves -- 2.1.6 The Complex Line Integral -- 2.1.7 Properties of Integrals -- 2.2 Complex Differentiabilityand Conformality -- 2.2.1 Limits -- 2.2.2 Holomorphicity and the Complex Derivative -- 2.2.3 Conformality -- 2.3 The Cauchy Integral Formula and Theorem -- 2.3.1 The Cauchy Integral Theorem, Basic Form -- 2.3.2 The Cauchy Integral Formula -- 2.3.3 More General Forms of the Cauchy Theorems -- 2.3.4 Deformability of Curves -- 2.4 A Coda on the Limitations of The Cauchy Integral Formula -- 3 Applications of the Cauchy Theory -- 3.1 The Derivatives of a Holomorphic Function -- 3.1.1 A Formula for the Derivative.

3.1.2 The Cauchy Estimates -- 3.1.3 Entire Functions and Liouville's Theorem -- 3.1.4 The Fundamental Theorem of Algebra -- 3.1.5 Sequences of Holomorphic Functions and their Derivatives -- 3.1.6 The Power Series Representation of a Holomorphic Function -- 3.2 The Zeros of a Holomorphic Function -- 3.2.1 The Zero Set of a Holomorphic Function -- 3.2.2 Discreteness of the Zeros of a Holomorphic Function -- 3.2.3 Discrete Sets and Zero Sets -- 3.2.4 Uniqueness of Analytic Continuation -- 4 Isolated Singularities and Laurent Series -- 4.1 The Behavior of a Holomorphic Function near an Isolated Singularity -- 4.1.1 Isolated Singularities -- 4.1.2 A Holomorphic Function on a Punctured Domain -- 4.1.3 Classification of Singularities -- 4.1.4 Removable Singularities, Poles, and Essential Singularities -- 4.1.5 The Riemann Removable Singularities Theorem -- 4.1.6 The Casorati-Weierstrass Theorem -- 4.2 Expansion around Singular Points -- 4.2.1 Laurent Series -- 4.2.2 Convergence of a Doubly Infinite Series -- 4.2.3 Annulus of Convergence -- 4.2.4 Uniqueness of the Laurent Expansion -- 4.2.5 The Cauchy Integral Formula for an Annulus -- 4.2.6 Existence of Laurent Expansions -- 4.2.7 Holomorphic Functions with Isolated Singularities -- 4.2.8 Classification of Singularities in Terms of Laurent Series -- 4.3 Examples of Laurent Expansions -- 4.3.1 Principal Part of a Function -- 4.3.2 Algorithm for Calculating the Coefficients of the Laurent Expansion -- 4.4 The Calculus of Residues -- 4.4.1 Functions with Multiple Singularities -- 4.4.2 The Residue Theorem -- 4.4.3 Residues -- 4.4.4 The Index or Winding Number of a Curve about a Point -- 4.4.5 Restatement of the Residue Theorem -- 4.4.6 Method for Calculating Residues -- 4.4.7 Summary Charts of Laurent Series and Residues -- 4.5 Applications to the Calculation of Definite Integrals and Sums.

4.5.1 The Evaluation of Definite Integrals -- 4.5.2 A Basic Example of the Indefinite Integral -- 4.5.3 Complexification of the Integrand -- 4.5.4 An Example with a More Subtle Choice of Contour -- 4.5.5 Making the Spurious Part of the Integral Disappear -- 4.5.6 The Use of the Logarithm -- 4.5.7 Summing a Series Using Residues -- 4.6 Singularities at Infinity -- 4.6.1 Meromorphic Functions -- 4.6.2 Definition of Meromorphic Function -- 4.6.3 Examples of Meromorphic Functions -- 4.6.4 Meromorphic Functions with Infinitely Many Poles -- 4.6.5 Singularities at Infinity -- 4.6.6 The Laurent Expansion at Infinity -- 4.6.7 Meromorphic at Infinity -- 4.6.8 Meromorphic Functions in the Extended Plane -- 5 The Argument Principle -- 5.1 Counting Zeros and Poles -- 5.1.1 Local Geometric Behavior of a Holomorphic Function -- 5.1.2 Locating the Zeros of a Holomorphic Function -- 5.1.3 Zero of Order n -- 5.1.4 Counting the Zeros of a Holomorphic Function -- 5.1.5 The Argument Principle -- 5.1.6 Location of Poles -- 5.1.7 The Argument Principle for Meromorphic Functions -- 5.2 The Local Geometry of Holomorphic Functions -- 5.2.1 The Open Mapping Theorem -- 5.3 Further Results on the Zeros of Holomorphic Functions -- 5.3.1 Rouche's Theorem -- 5.3.2 Typical Application of Rouche's Theorem -- 5.3.3 Rouche's Theorem and the Fundamental Theorem of Algebra -- 5.3.4 Hurwitz's Theorem -- 5.4 The Maximum Principle -- 5.4.1 The Maximum Modulus Principle -- 5.4.2 Boundary Maximum Modulus Theorem -- 5.4.3 The Minimum Principle -- 5.4.4 The Maximum Principle on an Unbounded Domain -- 5.5 The Schwarz Lemma -- 5.5.1 Schwarz's Lemma -- 5.5.2 The Schwarz-Pick Lemma -- 6 The Geometric Theory of Holomorphic Functions -- 6.1 The Idea of a Conformal Mapping -- 6.1.1 Conformal Mappings -- 6.1.2 Conformal Self-Maps of the Plane -- 6.2 Linear Fractional Transformations.

6.2.1 Linear Fractional Mappings -- 6.2.2 The Topology of the Extended Plane -- 6.2.3 The Riemann Sphere -- 6.2.4 Conformal Self-Maps of the Riemann Sphere -- 6.2.5 The Cayley Transform -- 6.2.6 Generalized Circles and Lines -- 6.2.7 The Cayley Transform Revisited -- 6.2.8 Summary Chart of Linear Fractional Transformations -- 6.3 The Riemann Mapping Theorem -- 6.3.1 The Concept of Homeomorphism -- 6.3.2 The Riemann Mapping Theorem -- 6.3.3 The Riemann Mapping Theorem: Second Formulation -- 6.4 Conformal Mappings of Annuli -- 6.4.1 A Riemann Mapping Theorem for Annuli -- 6.4.2 Conformal Equivalence of Annuli -- 6.4.3 Classification of Planar Domains -- 7 Harmonic Functions -- 7.1 Basic Properties of Harmonic Functions -- 7.1.1 The Laplace Equation -- 7.1.2 Definition of Harmonic Function -- 7.1.3 Real- and Complex-Valued Harmonic Functions -- 7.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions -- 7.1.5 Smoothness of Harmonic Functions -- 7.2 The Maximum Principle and the Mean Value Property -- 7.2.1 The Maximum Principle for Harmonic Functions -- 7.2.2 The Minimum Principle for Harmonic Functions -- 7.2.3 The Boundary Maximum and Minimum Principles -- 7.2.5 Boundary Uniqueness for Harmonic Functions -- 7.3 The Poisson Integral Formula -- 7.3.1 The Poisson Integral -- 7.3.2 The Poisson Kernel -- 7.3.3 The Dirichlet Problem -- 7.3.4 The Solution of the Dirichlet Problem on the Disc -- 7.3.5 The Dirichlet Problem on a General Disc -- 7.4 Regularity of Harmonic Functions -- 7.4.1 The Mean Value Property on Circles -- 7.4.2 The Limit of a Sequence of Harmonic Functions -- 7.5 The Schwarz Reflection Principle -- 7.5.1 Reflection of Harmonic Functions -- 7.5.2 Schwarz Reflection Principle for Harmonic Functions -- 7.5.3 The Schwarz Reflection Principle for Holomorphic Functions.

7.5.4 More General Versions of the Schwarz Reflection Principle -- 7.6 Harnack's Principle -- 7.6.1 The Harnack Inequality -- 7.6.2 Harnack's Principle -- 7.7 The Dirichlet Problem and Subharmonic Functions -- 7.7.1 The Dirichlet Problem -- 7.7.2 Conditions for Solving the Dirichlet Problem -- 7.7.4 Definition of Subharmonic Function -- 7.7.5 Other Characterizations of Subharmonic Functions -- 7.7.6 The Maximum Principle -- 7.7.7 Lack of A Minimum Principle -- 7.7.8 Basic Properties of Subharmonic Functions -- 7.7.9 The Concept of a Barrier -- 7.8 The General Solution of the Dirichlet Problem -- 7.8.1 Enunciation of the Solution of the Dirichlet Problem -- 8 Infinite Series and Products -- 8.1 Basic Concepts Concerning Infinite Sums and Products -- 8.1.1 Uniform Convergence of a Sequence -- 8.1.2 The Cauchy Condition for a Sequence of Functions -- 8.1.3 Normal Convergence of a Sequence -- 8.1.4 Normal Convergence of a Series -- 8.1.5 The Cauchy Condition for a Series -- 8.1.6 The Concept of an Infinite Product -- 8.1.7 Infinite Products of Scalars -- 8.1.8 Partial Products -- 8.1.9 Convergence of an Infinite Product -- 8.1.10 The Value of an Infinite Product -- 8.1.11 Products That Are Disallowed -- 8.1.12 Condition for Convergence of an Infinite Product -- 8.1.13 Infinite Products of Holomorphic Functions -- 8.1.14 Vanishing of an Infinite Product -- 8.1.15 Uniform Convergence of an Infinite Product of Functions -- 8.1.16 Condition for the Uniform Convergence of an Infinite Product of Functions -- 8.2 The WeierstrassFactorization Theorem -- 8.2.1 Prologue -- 8.2.2 Weierstrass Factors -- 8.2.3 Convergence of the Weierstrass Product -- 8.2.4 Existence of an Entire Function with Prescribed Zeros -- 8.2.5 The Weierstrass Factorization Theorem -- 8.3 The Theorems of Weierstrass and Mittag-Leffler -- 8.3.1 The Concept of Weierstrass's Theorem.

8.3.2 Weierstrass's Theorem.