Theory of  Functions of a Complex Variable


Meets in HILL 423   TTh 5 (2:50-4:10)
 

Ovidiu Costin
office: H214
tel. 732 445 2472
costin@math.rutgers.edu

The course covers: elementary properties of complex numbers, analytic
functions, the Cauchy-Riemann equations, power series, Cauchy's Theorem,
zeros and singularities of analytic functions, maximum modulus
principle, conformal mapping, Schwarz's lemma, the residue theorem,
Schwarz's reflection principle, the argument principle, Rouché's
theorem, normal families, the Riemann mapping theorem, properties of
meromorphic functions, the Phragmen-Lindelof principle and elementary
properties of harmonic functions.

Text: Serge Lang, Complex Analysis , 4th edition.

Prerequisite: Advanced calculus.

Approximate syllabus:

  1. × The algebra of complex numbers and complex valued functions.
  2. × Elementary topology of the plane.
  3. × Complex differentiability; the Cauchy-Riemann equations; angles under holomorphic maps.
  4. × Power series, operations with power series.
  5. × Convergence criteria, radius of convergence, Abel's theorem.
  6. × Inverse mapping theorem, open mapping theorem, local maximum modulus principle.
  7. × Holomorphic functions on connected sets. Elementary analytic continuation.
  8. × Integrals over paths.
  9. × Primitive of a holomorphic function. The Cauchy-Goursat theorem.
  10. × Integrals along continuous curves, homotopy form of Cauchy's theorem.
  11. × Global primitives, definition of the logarithm.
  12. × Local Cauchy formula, Liouville's theorem, Cauchy estimates, Morera's theorem.
  13. × Winding number, global Cauchy theorem.
  14. × Uniform limits, isolated singularities.
  15. × Laurent series.
  16. ×The residue formula.
  17. × Evaluation of definite integrals using the residue theorem.
  18. × More calculations with the residue theorem.
  19. × Conformal mapping, Schwarz lemma, automorphisms of the disc and upper half plane.
  20. × Other examples of conformal mappings. Level sets.
  21. × Fractional linear transformations.
  22. × Harmonic functions.
  23. × More properties of harmonic functions, the Poisson formula.
  24. × Normal families, formulation and proof of the Riemann Mapping Theorem.
  25. × Weierstrass products. Functions of finite order. Minimum modulus principle.
  26. × Meromorphic functions, the Mittag-Lefler theorem
  27. The Phragmen-Lindelof principle.
  28. The D-bar operator.

 
Homework problems to be turned in :

Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7    Set 8    Set 9