Wenhan Dai

Complex Analysis for the Generals

Instruction Watch: My notes for complex analysis are available now. These notes are intended to be written for the PhD Entrance and Qualifying Examination at the Beijing International Center of Mathematical Research in 2022. The first previous chapters do follow [SS10] loosely so that I claim no originality. Another standard references for these notes are [Lan03] and [Kod07]. The outline is as follows.

Outline

See the most recent notes.

  1. Holomorphic Functions.
    Cauchy-Riemann Equations; Cauchy Theorem of local and global versions: the existence of primitives; Cauchy Integral Formula; Holomorphicity is equivalent to analyticity; The existence of complex logarithm on simply connected regions; Liouville Theorem: the rigidity of entire functions; Montel Theorem; The Mean-Value Property; The Maximum Principle; Open Mapping Theorem.
  2. Meromorphic Functions.
    Zeros and poles, local expansion near zeros and poles; The Residue Formula; Application I: evaluation of integrals; Application II: the argument principle; Rouché Theorem.
  3. On Fourier Transform.
    Poisson Summation Formula; Paley-Wiener Theorem.
  4. Entire Functions.
    Jensen’s Formula; Weierstrass infinite products; Hadamard Factorization Theorem; Basics of Nevanlinna Theory.
  5. Special Functions.
    Analytic continuation of Gamma function; Symmetry and other properties of Gamma function; Zeta function and Xi function.
  6. The Prime Number Theory.
    Euler Identity; Locations of Zeros of Zeta function; The Prime Number Theorem.
  7. Geometric Theory of Holomorphic Functions.
    Conformal/biholomorphic maps; The unit disc is conformally equivalent to the upper-half plane; Schwarz Lemma: to compute Aut(D) and Aut(H); D is a hyperbolic space; The Riemann Mapping Theorem; Boundary correspondences and the construction of a modular function.
  8. Ellptic Functions.
    Weierstrass p-function on lattices and the elliptic curve; Fourier transform and q-expansion; The SL(2,Z)-action and its fundamental domain.
  9. The Theta Function.
    The Triple-Product Formula; Applications to combinatorics and number theory.

References

  1. [Kod07] Kunihiko Kodaira. Complex analysis, volume 107. Cambridge University Press, 2007.
  2. [Lan03] Serge Lang. Complex analysis, volume 103. Springer Science & Business Media, 2003.
  3. [SS10] Elias M. Stein and Rami Shakarchi. Complex analysis, volume 2. Princeton University Press, 2010.