Math Ladder · College

Complex Analysis

Calculus with complex numbers: analytic functions, the complex exponential and logarithm, contour integrals, Cauchy's formula, Laurent series, residues and conformal maps.

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42 topics in 8 units. Builds on Calculus 3: Multivariable Calculus and Calculus 4: Vector Calculus. Not sure this is the right course? Find your rung.

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Unit 1 Complex Numbers and the Plane

4 topics

Work with modulus, conjugate, exponential form and arguments, and describe regions of the plane.

  1. Modulus and Conjugate
  2. Euler's Formula and Exponential Form
  3. Arguments and the Principal Argument
  4. Curves and Regions in the Plane

Unit 2 Complex Functions

3 topics

Split a complex function into real and imaginary parts, see it as a map of the plane, and take limits from every direction.

  1. Complex Functions as u + iv
  2. Functions as Mappings
  3. Limits and Continuity

Unit 3 Analytic Functions

4 topics

Differentiate in the complex sense, test with the Cauchy–Riemann equations, and meet harmonic functions.

  1. The Complex Derivative
  2. The Cauchy–Riemann Equations
  3. Analytic Functions
  4. Harmonic Functions and Harmonic Conjugates

Unit 4 Exponentials, Logarithms and Powers

5 topics

Extend eˣ, sine, cosine, logarithms and powers to complex numbers, and handle the many values of log z with branches.

  1. The Complex Exponential
  2. Complex Sine and Cosine
  3. The Complex Logarithm
  4. Branches and Branch Cuts
  5. Complex Powers

Unit 5 Contour Integrals and Cauchy's Theorem

10 topics

Integrate along curves in the plane, then use Cauchy's theorem and integral formula and their consequences.

  1. Contours and Parametrizations
  2. Contour Integrals
  3. Bounding Integrals: The ML Inequality
  4. Antiderivatives and Path Independence
  5. Cauchy's Theorem
  6. Deforming Contours
  7. Cauchy's Integral Formula
  8. Cauchy's Formula for Derivatives
  9. Liouville's Theorem and the Fundamental Theorem of Algebra
  10. The Maximum Modulus Principle

Unit 6 Power Series and Laurent Series

4 topics

Expand analytic functions in power series on disks and Laurent series on annuli, and classify singularities.

  1. Power Series and Disks of Convergence
  2. Taylor Series of Analytic Functions
  3. Laurent Series
  4. Zeros and Isolated Singularities

Unit 7 Residues

8 topics

Compute residues, use the residue theorem to evaluate real integrals, and count zeros with the argument principle.

  1. Residues at Simple Poles
  2. Residues at Higher-Order Poles
  3. The Residue Theorem
  4. Integrals over a Full Turn
  5. Improper Integrals with Residues
  6. Integrals with Sines and Cosines
  7. The Argument Principle
  8. Rouché's Theorem

Unit 8 Conformal Maps

4 topics

Use analytic functions to map regions onto one another while keeping angles, with Möbius transformations as the main tool.

  1. Conformal Maps
  2. Möbius Transformations
  3. A Möbius Map from Three Points
  4. Mapping Standard Regions
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