Introduction to Complex Analysis

These notes are based on lectures given in Math 196 at UNC. The core idea of complex analysis is that all the basic functions that arise in calculus, first derived as functions of a real variable, such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, and a host of more sophisticated functions, are actually naturally defined for complex arguments, and are complex-differentiable (a.k.a. holomorphic). Furthermore, the study of these functions on the complex plane reveals their structure more truly and deeply than one could imagine by only thinking of them as defined for real arguments.

Contents

  1. Holomorphic functions: definition and basic properties
  2. Holomorphic functions defined by power series
  3. Exponential and trigonometric functions: Euler's formula
  4. Square roots, logs, and other inverse functions
  5. The Cauchy integral theorem and the Cauchy integral formula
  6. Harmonic functions and the maximum principle
  7. Liouville's theorem and the fundamental theorem of algebra
  8. Morera's theorem and the Schwarz reflection principle
  9. Goursat's theorem
  10. Uniqueness of holomorphic functions
  11. Singularities
  12. Laurent series
  13. Fourier series and the Poisson integral
  14. Fourier transforms
  15. Laplace transforms
  16. Residue calculus
  17. The argument principle
  18. The Gamma function
  19. The Riemann zeta function
  20. Covering maps and inverse functions
  21. Normal families
  22. Conformal maps
  23. The Riemann mapping theorem
  24. Boundary behavior of conformal maps
  25. The disk covers C \ {0,1}
  26. The Riemann sphere and other Riemann surfaces
  27. Montel's theorem
  28. Picard's theorems
  29. Harnack estimates and more Liouville theorems
  30. Periodic and doubly periodic functions - infinite series representations
  31. The Weierstrass P in elliptic function theory
  32. Theta functions and P
  33. Elliptic integrals
  34. The Riemann surface of sqrt q(z)