• 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
• Forms of the Cauchy-Riemann equations
• Geometry of the Cauchy-Riemann equations

• Mappings lecture notes

• Integration lecture notes

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

• The ratio test for power series
• 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
• Symmetry of the gamma function
• Fourier inversion (and illustrating Mathematica graphics)
• The Heisenberg uncertainty principle
• The Bernoulli numbers, power sums, and zeta values
• A series representation of the cotangent
• The zeta function at negative odd integers, a la Euler

• 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
• SU(2)->SO(3)

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

• A basic theta function
• Continuations and functional equations
• Sum of logarithms, logarithm of product
• Primes in arithmetic progressions
• The Prime Number Theorem
• Weierstrass/Hadamard products

• Complex tori
• Complex tori as elliptic curves
• Sketch of function theory on complex tori
• A convergence criterion for infinite products
• Addition and duplication laws for Weierstrass P
• A quick introduction to elliptic curves
• 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)
• Beginning compact Riemann surface theory

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

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