## 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.