Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of results. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

The catalog description for the class Complex Variables (MATH 4337/5337) is: "Complex numbers and their algebra, complex differentiation and integration, analytic and elementary functions, residues and power series."

1. Chapter 1. Complex Numbers.
2. Chapter 2. Analytic Functions.
3. Chapter 3. Elementary Functions.
4. Chapter 4. Integrals.
5. Chapter 5. Series.
6. Chapter 6. Residues and Poles.

Chapter 1. Complex Numbers.

• Section 1. Sums and Products. Section 1.1 notes
• Section 2. Basic Algebraic Properties. Section 1.2 notes
• Supplement. Proofs of Theorems in Section 1.2. Beamer file of Section 1.2 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.2. Print file of Section 1.2 proofs and examples
• Section 3. Further Properties. Section 1.3 notes
• Supplement. Proofs of Theorems in Section 1.3. Beamer file of Section 1.3 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.3. Print file of Section 1.3 proofs and examples
• Section 4. Vectors and Moduli. Section 1.4 notes
• Supplement. Proofs of Theorems in Section 1.4. Beamer file of Section 1.4 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.4. Print file of Section 1.4 proofs and examples
• Section 5. Complex Conjugates. Section 1.5 notes
• Supplement. Proofs of Theorems in Section 1.5. Beamer file of Section 1.5 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.5. Print file of Section 1.5 proofs and examples
• Section 6. Exponential Form. Section 1.6 notes (This section includes the definition of argument and principal argument.)
• Section 7. Products and Powers in Exponential Form. Section 1.7 notes
• Supplement. Proofs of Theorems in Section 1.7. Beamer file of Section 1.7 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.7. Print file of Section 1.7 proofs and examples
• Section 8. Arguments of Products and Quotients. Section 1.8 notes
• Supplement. Proofs of Theorems in Section 1.8. Beamer file of Section 1.8 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.8. Print file of Section 1.8 proofs and examples
• Section 9. Roots of Complex Numbers. Section 1.9 notes
• Section 10. Examples. Section 1.10 notes
• Section 11. Regions in the Complex Plane. Section 1.11 notes
• Supplement. Proofs of Theorems in Section 1.11. Beamer file of Section 1.11 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 1.11. Print file of Section 1.11 proofs and examples
• Study Guide 1

Chapter 2. Analytic Functions.

• Section 12. Functions of a Complex Variable. Section 2.12 notes
• Section 13. Mappings. Section 2.13 notes
• Section 14. Mappings of the Exponential Function. Section 2.14 notes
• Section 15. Limits. Section 2.15 notes
• Supplement. Proofs of Theorems in Section 2.15. Beamer file of Section 2.15 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.15. Print file of Section 2.15 proofs and examples
• Section 16. Theorems on Limits. Section 2.16 notes
• Supplement. Proofs of Theorems in Section 2.16. Beamer file of Section 2.16 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.16. Print file of Section 2.16 proofs and examples
• Section 17. Limits Involving the Point at Infinity. Section 2.17 notes
• Supplement. Proofs of Theorems in Section 2.17. Beamer file of Section 2.17 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.17. Print file of Section 2.17 proofs and examples
• Section 18. Continuity. Section 2.18 notes
• Supplement. Proofs of Theorems in Section 2.18. Beamer file of Section 2.18 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.18. Print file of Section 2.18 proofs and examples
• Section 19. Derivatives. Section 2.19 notes
• Supplement. Proofs of Theorems in Section 2.19. Beamer file of Section 2.19 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.19. Print file of Section 2.19 proofs and examples
• Section 20. Differentiation Formulas. Section 2.20 notes
• Supplement. Proofs of Theorems in Section 2.20. Beamer file of Section 2.20 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.20. Print file of Section 2.20 proofs and examples
• Section 21. Cauchy-Riemann Equations. Section 2.21 notes
• Supplement. Proofs of Theorems in Section 2.21. Beamer file of Section 2.21 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.21. Print file of Section 2.21 proofs and examples
• Section 22. Sufficient Conditions for Differentiability. Section 2.22 notes
• Supplement. Proofs of Theorems in Section 2.22. Beamer file of Section 2.22 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.22. Print file of Section 2.22 proofs and examples
• Section 23. Polar Coordinates. Section 2.23 notes
• Supplement. Proofs of Theorems in Section 2.23. Beamer file of Section 2.23 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.23. Print file of Section 2.23 proofs and examples
• Section 24. Analytic Functions. Section 2.24 notes
• Supplement. Proofs of Theorems in Section 2.24. Beamer file of Section 2.24 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.24. Print file of Section 2.24 proofs and examples
• Section 25. Examples. Section 2.25 notes
• Supplement. Proofs of Theorems in Section 2.25. Beamer file of Section 2.25 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.25. Print file of Section 2.25 proofs and examples
• Section 26. Harmonic Functions. Section 2.26 notes
• Supplement. Proofs of Theorems in Section 2.26. Beamer file of Section 2.26 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.26. Print file of Section 2.26 proofs and examples
• Section 27. Uniquely Determined Analytic Functions. Section 2.27 notes
• Supplement. Proofs of Theorems in Section 2.27. Beamer file of Section 2.27 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.27. Print file of Section 2.27 proofs and examples
• Section 28. Reflection Principle. Section 2.28 notes
• Supplement. Proofs of Theorems in Section 2.28. Beamer file of Section 2.28 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 2.28. Print file of Section 2.28 proofs and examples
• Study Guide 2

Chapter 3. Elementary Functions.

• Section 29. The Exponential Function. Section 3.29 notes
• Supplement. Proofs of Theorems in Section 3.29. Beamer file of Section 3.29 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 3.29. Print file of Section 3.29 proofs and examples
• Section 30. The Logarithm Function. Section 3.30 notes
• Section 31. Branches and Derivatives of Logarithms. Section 3.31 notes
• Section 32. Some Identities Involving Logarithms. Section 3.32 notes
• Supplement. Proofs of Theorems in Section 3.32. Beamer file of Section 3.32 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 3.32. Print file of Section 3.32 proofs and examples
• Section 33. Complex Exponents. Section 3.33 notes
• Supplement. Proofs of Theorems in Section 3.33. Beamer file of Section 3.33 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 3.33. Print file of Section 3.33 proofs and examples
• Section 34. Trigonometric Functions. Section 3.34 notes
• Supplement. Proofs of Theorems in Section 3.34. Beamer file of Section 3.34 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 3.34. Print file of Section 3.34 proofs and examples
• Section 35. Hyperbolic Functions. Section 3.35 notes
• Section 36. Inverse Trigonometric and Hyperbolic Functions. Section 3.36 notes
• Study Guide 3

Chapter 4. Integrals.

• Section 37. Derivatives of Functions w(t). Section 4.37 notes
• Section 38. Definite Integrals of Functions w(t). Section 4.38 notes
• Section 39. Contours. Section 4.39 notes
• Section 40. Contour Integrals. Section 4.40 notes
• Section 41. Some Examples. Section 4.41 notes
• Section 42. Examples with Branch Cuts. Section 4.42 notes
• Section 43. Upper Bounds for Moduli of Contour Integrals. Section 4.43 notes
• Supplement. Proofs of Theorems in Section 4.43. Beamer file of Section 4.43 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.43. Print file of Section 4.43 proofs and examples
• Section 44. Antiderivatives. Section 4.44 notes
• Section 45. Proof of the Theorem. Section 4.45 notes
• Section 46. Cauchy-Goursat Theorem. Section 4.46 notes
• Section 47. Proof of the Theorem. Section 4.47 notes
• Supplement. Proofs of Theorems in Section 4.47. Beamer file of Section 4.47 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.47. Print file of Section 4.47 proofs and examples
• Section 48. Simply Connected Domains. Section 4.48 notes
• Supplement. Proofs of Theorems in Section 4.48. Beamer file of Section 4.48 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.48. Print file of Section 4.48 proofs and examples
• Section 49. Multiply Connected Domains. Section 4.49 notes
• Supplement. Proofs of Theorems in Section 4.49. Beamer file of Section 4.49 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.49. Print file of Section 4.49 proofs and examples
• Section 50. Cauchy Integral Formula. Section 4.50 notes
• Supplement. Proofs of Theorems in Section 4.50. Beamer file of Section 4.50 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.50. Print file of Section 4.50 proofs and examples
• Section 51. An Extension of the Cauchy Integral Formula. Section 4.51 notes
• Supplement. Proofs of Theorems in Section 4.51. Beamer file of Section 4.51 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.51. Print file of Section 4.51 proofs and examples
• Section 52. Some Consequences of the Extension. Section 4.52 notes
• Supplement. Proofs of Theorems in Section 4.52. Beamer file of Section 4.52 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.52. Print file of Section 4.52 proofs and examples
• Section 53. Liouville's Theorem and the Fundamental Theorem of Algebra. Section 4.53 notes (Contains a proof of the Fundamental Theorem of Algebra.)
• Supplement. Proofs of Theorems in Section 4.53. Beamer file of Section 4.53 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.53. Print file of Section 4.53 proofs and examples
• Section 54. Maximum Modulus Principle. Section 4.54 notes
• Supplement. Proofs of Theorems in Section 4.54. Beamer file of Section 4.54 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 4.54. Print file of Section 4.54 proofs and examples
• Supplement. Applications of the Maximum Modulus Theorem to Polynomials. Applications of the Maximum Modulus Theorem notes
• Supplement. Proofs of Theorems in Supplement to Section 54. Beamer file of Applications of the Maximum Modulus Theorem proofs
• Supplement. Printout of the Proofs of Theorems in Supplement to Section 54. Print file of Proofs of Theorems in Supplement to Section 4.54
• Study Guide 4

Chapter 5. Series.

• Section 55. Convergence of Sequences. Section 5.55 notes
• Supplement. Proofs of Theorems in Section 5.55. Beamer file of Section 5.55 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.55. Print file of Section 5.55 proofs and examples
• Section 56. Convergence of Series. Section 5.56 notes
• Supplement. Proofs of Theorems in Section 5.56. Beamer file of Section 5.56 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.56. Print file of Section 5.56 proofs and examples
• Section 57. Taylor Series. Section 5.57 notes (Contains a claim that an analytic function has a power series representation.)
• Section 58. Proof of Taylor's Theorem. Section 5.58 notes
• Supplement. Proofs of Theorems in Section 5.58. Beamer file of Section 5.58 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.58. Print file of Section 5.58 proofs and examples
• Section 59. Examples. Section 5.59 notes
• Section 60. Laurent Series. Section 5.60 notes
• Section 61. Proof of Laurent's Theorem. Section 5.61 notes
• Section 62. Examples. Section 5.62 notes
• Section 63. Absolute and Uniform Convergence of Power Series. Section 5.63 notes
• Supplement. Proofs of Theorems in Section 5.63. Beamer file of Section 5.63 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.63. Print file of Section 5.63 proofs and examples
• Section 64. Continuity of Sums of Power Series. Section 5.64 notes
• Supplement. Proofs of Theorems in Section 5.64. Beamer file of Section 5.64 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.64. Print file of Section 5.64 proofs and examples
• Section 65. Integration and Differentiation of Power Series. Section 5.65 notes
• Supplement. Proofs of Theorems in Section 5.65. Beamer file of Section 5.65 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.65. Print file of Section 5.65 proofs and examples
• Section 66. Uniqueness of Series Representations. Section 5.66 notes
• Supplement. Proofs of Theorems in Section 5.66. Beamer file of Section 5.66 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 5.66. Print file of Section 5.66 proofs and examples
• Section 67. Multiplication and Division of Power Series. Section 5.67 notes
• Study Guide 5

Chapter 6. Residues and Poles.

• Section 68. Isolated Singular Points. Section 6.68 notes
• Section 69. Residues. Section 6.69 notes
• Section 70. Cauchy's Residue Theorem. Section 6.70 notes
• Supplement. Proofs of Theorems in Section 6.70. Beamer file of Section 6.70 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 6.70. Print file of Section 6.70 proofs and examples
• Section 71. Residues at Infinity. Section 6.71 notes
• Supplement. Proofs of Theorems in Section 6.71. Beamer file of Section 6.71 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 6.71. Print file of Section 6.71 proofs and examples
• Section 72. The Three Types of Isolated Singular Points. Section 6.72 notes
• Section 73. Residues at Poles. Section 6.73 notes
• Supplement. Proofs of Theorems in Section 6.73. Beamer file of Section 6.73 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 6.73. Print file of Section 6.73 proofs and examples
• Section 74. Examples. Section 6.74 notes
• Section 75. Zeros of Analytic Functions. Section 6.75 notes
• Supplement. Proofs of Theorems in Section 6.75. Beamer file of Section 6.75 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 6.75. Print file of Section 6.75 proofs and examples
• Section 76. Zeros and Poles. Section 6.76 notes
• Supplement. Proofs of Theorems in Section 6.76. Beamer file of Section 6.76 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 6.76. Print file of Section 6.76 proofs and examples
• Section 77. Behavior of Functions Near Isolated Singular Points. Section 6.77 notes
• Supplement. Proofs of Theorems in Section 6.77. Beamer file of Section 6.77 proofs and examples
• Supplement. Printout of the Proofs of Theorems in Section 6.77. Print file of Section 6.77 proofs and examples
• Study Guide 6

Additional Chapters. A complete list of the sections in these chapters is also available.

• Chapter 7. Applications of Residues.
• Chapter 8. Mapping by Elemetary Functions.
• Chapter 9. Conformal Mapping.
• Chapter 10. Applications of Conformal Mapping.
• Chapter 11. The Schwarz-Christoffel Transformation.
• Chapter 12. Integral Formulas of the Poisson Type.

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