Stony Brook - Graduate Courses - Complex Analysis I - MAT 542
Complex Analysis I
MAT 542
Spring Semester
The field of complex numbers, geometric representation of complex numbers
Analytic functions
Definition, Cauchy-Riemann equations
Elementary theory of power series, uniform convergence
Elementary functions: rational, exponential and trigonometric functions
The logarithm
Analytic functions as mappings
Conformality
Linear fractional transformations
Elementary conformal mappings
Complex integration
Line integrals and Cauchy's theorem for disk and rectangle
Cauchy's integral formula
Cauchy's inequalities
Morera's theorem, Liouville's theorem and fundamental theorem of algebra
The general form of Cauchy's theorem
Local properties of analytic functions
Removable singularities, Taylor's theorem
Zeros and poles, classification of isolated singularities
The local mapping theorem
The maximum modulus principle, Schwarz's lemma
The calculus of residues
The residue theorem
The argument principle
Rouche's theorem
Evaluation of definite integrals
Power series
Weierstrass theorem
The Taylor and Laurent series
Partial fractions and infinite products
Normal families
The Riemann mapping theorem
Harmonic functions
The mean-value property
Harnack's inequality
The Dirichlet problem
Typical references:
Lars V. Ahlfors,
Complex analysis: an introduction to the theory of analytic functions of one complex variable,
3
^{rd}
ed.; McGraw-Hill, 1979;
John B. Conway,
Functions of one complex variable,
2
^{nd}
ed.; Springer-Verlag, 1978.