Complex Analysis (MA 412) by Prof. Gopal Krishna Srinivasan

-1st class

  • Prerequisites: Real Analysis, Fields. Set of Complex Numbers as a field. Field operations are compatible with metric space properties to make them continuous. Complex numbers as a field cannot be ordered. \(\mathbb{C}\) is a field which fails to be a factorization domain, but whenever factorization possible, it is unique. Recalled Integral domains and Unique factorization domains, primes and irreducibles. What are the primes and irreducibles in Complex field?
  • Starts with motivation of studying Complex Analysis and the fields it interacts with. Riemann’s dissertation, Cauchy’s approach, Poissin, Mertens. Use of complex analysis in Special functions in Number theory: Riemann Zeta function and the resulting Riemann Hypothesis. Prime number theorem.
  • Electrostatics anf Complex Analysis. Cauchy’s Integral Theorem as the 2D version of Gauss’ Divergence theorem. Potential Theory techniques
  • Algebraic Topology and Complex Ananlysis- Jordan Curve Theorem, its generalization: the Schoenflies theorem in 2D, not true in higher dimensions- special topology of the plane. Lifting of holomorphic functions from unit disc to entire complex plane.
  • Riemann Mapping Theorem via PDEs approach. Solve Laplace Equation with boundary value specified- Dirichlet Problem. Used characterization of solution by minimizing the energy integral, existence of minimizer not proved but based on physical grounds (electrostatics)- Dirichlet Principle.

-Later in the day, second class

  • Complex differentiability and Cauchy Riemann Equations. Holomorphic functions and Harmonic Conjugates, Properties of holomorphic functions-Conformal maps, non-constant holo take all but one value
  • Degree Theory: Critical values, regular values, Morse–Sard theorem, holomorphic functions have Jacobian non-negative, finitely many critical points, degree of a map.

Class 3

Class 4

Class 5

  • expz, sinz and cosz are entire. Little Picard theorem- non constant entire functions must take all complex values except 1. Sinx takes all values because if w is missed, -w is also missed (odd function)

Class 6

  • Sequences and series, Dirichlet’s theorem- absolutely convergent series limit doesn’t change on rearrangement of terms.
  • Power series and tests: comparison test, ratio test, root test
  • Union of open discs of convergence.
  • Radius of convergence
  • Gauss’s memoir on hypergeometric series and delicate test for convergence at boundary points.
  • Open + locally path connected = connected