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| Doctorate Complex Analysis Syllabus |
- Complex Numbers
- The field
- Geometry, linear fractional (Möbius) transformations, Riemann sphere
- Complex functions, analytic functions, Cauchy-Riemann equations
- Power series, exponential and trigonometric functions
- Conformality
- Cauchy Theory
- Line integrals
- Cauchy's theorem and formula, residues, singularities, calculation of integrals, maximum modulus principle
- Taylor and Laurent series
- Liouville's theorem, fundamental theorem of algebra, open mapping theorem, Rouche's formula
- Schwarz's Lemma, Jensen's formula, Weierstrass' theorem
- Representation Theorems
- Partial fractions
- Infinite products, entire functions, Hadamard's theorem
- Theorems of Mittag-Leffler, Weierstrass & Runge
- Harmonicity
- Harmonic functions, reflection principle
- Poisson integral
- Dirichlet problem
- Special Functions
- Gamma function, Riemann function
- Miscellaneous
- Normal families, Riemann mapping theorem
- Analytic continuation, monodromy theorem
- Picard's theorem
| References |
Conway: Functions of One Complex Variable
Ahlfors: Complex Analysis
Rudin: Real and Complex Analysis
Hille: Analytic Function Theory
Heins: Complex Function Theory
Churchill: Complex Variables and Applications
Veech: A Second Course in Complex Variables
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