## Lecture Notes (pdf format)

- Lecture 0
(Definition of complex numbers, polar coordinates, roots of unity)
- Lecture 1
(notion of complex differentiability, Cauchy-Riemann equations)
- Lecture 2
(Some proofs: of chain rule and product rule)
- Lecture 3
(complex integration, Cauchy's theorem for rectangles)
- Lecture 4
(Morera's Theorem, Cauchy's theorem in general)
- Lecture 5
(Liouville's Theorem, Fundamental theorem of algebra)
- Optional Reading 1
(some real analysis, proof of zig-zag paths and absolute max/min)
- Lecture 6
(rational functions, partial fractions)
- Lecture 7
(uniformly convergent sequences)
- Lecture 8
(Abel's Theorem, uniform convergence of power series)
- Lecture 9
(Identity Theorem)
- Lecture 10
(Classification of singularities, Residues)
- Lecture 11
(Singularity at infinity, functions of type z^a)
- Lecture 12
(Applications of residue theorem to real (infinite) integrals)
- Lecture 13
(Principal Value, more examples)
- Lecture 14
(Functions defined by integrals)
- Lecture 15
(Gamma function I: Euler's integral)
- Lecture 16
(Gamma function II: Weierstrass' infinite product)
- Lecture 17
(Gamma function III: Weierstrass = Euler)
- Lecture 18
(Gamma function IV: Stirling series)
- Lecture 19
(Elliptic functions: basics, definition of theta function)
- Lecture 20
(Theta function II. Jacobi's product formula)
- Lecture 21
(Theta function III. Fay's trisecant identity)
- Lecture 22
(Theta function IV. Jacobi's imaginary transform)