• Course introduction

• Preview lecture notes
• Exponential growth dominates polynomial growth

• Complex numbers lecture notes
• Stereographic projection is conformal
• Stereographic projection and the Mercator map

• Point set topology lecture notes
• Basic symbol-patterns for sets and maps
• Compactness and uniformity

• Complex differentiability and vector differentiability
• Sufficient conditions for complex differentiability
• Forms of the Cauchy-Riemann equations
• Geometry of the Cauchy-Riemann equations

• Mappings lecture notes

• Integration lecture notes
• A far-reaching little integral
• An existence theorem for the complex integral
• A compatibility verification for the complex integral
• Invariance of the complex integral
• The basic estimate for the complex integral
• A compatibility verification for path-length

• Cauchy's Theorem for simple curves
• Preview of three ideas
• Compactness and uniformity (again)
• Passing limits through integrals

• The radius of convergence formula
• Differentiability of power series
• Differentiability of power series, sans integrals
• Differentiability of power series via a more general result

• Consequences of power series representation
• Laurent series and singularities
• The Residue Theorem and its consequences

• Contour integrals lecture notes
• A gamma function identity
• A sketch of Fourier inversion
• The Bernoulli numbers and power sums
• A series representation of the cotangent
• The zeta function at negative odd integers, a la Euler
• Meromorphic continuation and functional equation of the Riemann zeta function
• Primes in arithmetic progressions

• Conformal mappings lecture notes
• Stereographic projection is conformal (again)
• A verification of associativity
• Circles and lines

• Automorphism groups lecture notes
• Rotations of the Riemann sphere

• The Arzela-Ascoli Theorem
• The Riemann Mapping Theorem
• Dirichlet's Problem on the disk

• Complex tori
• Complex tori as elliptic curves
• Sketch of function theory on complex tori
• Addition and duplication laws for Weierstrass P
• The elliptic curve group law via the Riemann-Roch Theorem
• Klein's j-function
• Sketch of Picard's Theorem

• Beginning modular forms
• Local coordinates on modular curves (figures)
• The genus of X_0(N) (Mathematica code)
• Beginning compact Riemann surface theory

Comments and corrections should be sent to jerry@reed.edu.

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