M2PM3 Complex Analysis 2018

Ari Laptev

 

Course information

The main goal of this course is to give an introduction to basic facts of Theory Complex functions.

Office hour: Fridays 13:00-14:00

Content

  • Complex numbers and the complex plane:Basic properties, Convergence, Sets in the complex plane.
  • Functions on the complex plane: Continuous functions, Holomorphic functions. Power series. Integration along curves.
  • Cauchy's Theorem and Its Applications: Goursat's theorem. Local existence of primitives and Cauchy's theorem in a disc. Evaluation of some integrals, Cauchy's integral formulae.
  • Meromorphic Functions: Zeros and poles. Laurent's Theorem. The residue formula. The argument principle and applications.
  • Conformal Mappings: Preservation of Angles, Mobius Transformations
  • Lectures

    Lecture 1 (ppt), Lecture 2 (ppt), Lecture 3 (ppt), Lecture 4&5 (ppt), Lecture 6 (ppt),

    Lecture 1 (pdf), Lecture 2 (pdf), Lecture 3 (pdf), Lecture 4&5 (pdf), Lecture 6 (pdf),

    Problems:

    probl.1 ( sol.1 ),

    Courseworks:

    The deadline for the first corsework is Tuesday, February 19 before 14:00.

    The deadline for the second corsework is Friday, March 22 before 14:00.

    Recommended Student Texts:

    Barry Simon, A Comprehensive Course in Analysis, Part 2A: Basic Complex Analysis, American Math Society, 2015.

    Elias M. Stein & Rami Shakarchi, II Complex Analysis, Princeton University Press, 2003.

    Elias M. Stein & Rami Shakarchi, I Fourier Analysis, Princeton University Press, 2003.

    John M. Howie, Complex Analysis, Springer, 2007.

    Walter Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974.

    Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979.