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Course Calendar  

Note:   This calendar is approximate and is subject to revision during the term.
  • Week 1 (August 7- August 11)       Sections 1 - 18
    • The Algebra of Complex Numbers.
    • The Geometry of Complex Numbers.
    • The Operations of Conjugation and Absolute Value, and Their Relationships with Sums and Products.
    • Polar Form and Exponential Form.
    • Roots of Complex Numbers.
    • Regions in the Complex Plane.
    • Functions of a Complex Variable.
    • Limits. Theorems on Limits.
    • Limits Involving Infinity. Riemann Sphere.
    • Continuity. Basic Properties of Continuous Functions.
  • Week 2 (August 14- August 18)       Sections 19 - 39
    • Differentiation. Basic Properties of and Formulas for Differentiation.
    • Cauchy-Riemann Equations.
    • Analytic Functions. Singular points.
    • Harmonic Functions. Harmonic Conjugates.
    • The Complex Exponential Function.
    • The Complex Trigonometric Functions.
    • Hyperbolic Functions.
    • The Complex Logarithm. Branch Cuts. Logarithmic Identities.
    • The Power Function.
  • Week 3 (August 21- August 25)       Sections 41 - 59
    • Calculus of Complex Functions of a Real Variable.
    • Arcs, Simple Closed Curves, and Contours.
    • Contour Integrals. Upper Bounds for Moduli of Contour Integrals.
    • Antiderivatives and Independence of Path.
    • Cauchy-Goursat Theorem. Simply Connected and Multiply Connected Domains. Principle of Deformation of Paths.
    • Cauchy Integral Formula. Generalized Cauchy Integral Formula.
    • Liouville's Theorem and the Fundamental Theorem of Algebra.
    • Maximum Modulus Principle.
  • Week 4 (August 28- September 1)       Sections 60 - 72
    • The Basics of Sequences and Series.
    • Taylor Series Representation of Analytic Functions. Maclaurin Series.
    • Laurent Series.
    • Uniform Convergence. Continuity of Power Series. Analyticity of Power Series.
  • Week 5 (September 4- September 8)       Sections 73 - 78
    • Multiplication and Division of Power Series.
    • Isolated Singular Points and Residues.
    • Cauchy's Residue Theorem.
    • Classification of Isolated Singularities.