Mathematics 4X03: Plan of Lectures (Fall, 2000)

Notes: Numbers in brackets refer to sections of the main and auxiliary textbooks.
GK - the book by Greene and Krantz; AF - the book by Ablowitz and Fokas.

1. Complex differentiation and holomorphic functions
• properties of complex numbers [GK: 1.1-1.2; AF: 1.1]
• complex polynomials and elementary functions [GK: 1.3; AF: 1.2.1]
• limits, continuity and complex differentiation [GK: 2.2; AF: 1.3]
• Cauchy--Riemann equations for holomorphic functions [GK: 1.4-1.5; AF: 2.1.1]
2. Complex integration and Cauchy integral formulas
• complex line integrals [GK: 2.1; AF: 2.4]
• Cauchy integral theorem [GK: 2.4-2.5; AF: 2.5]
• Cauchy integral formula [GK: 2.6,3.1; AF: 2.6.1]
• generalized Cauchy formula and D-bar derivative [GK: App.A; AF: 2.6.3]
3. Taylor and Laurent series
• Taylor series and convergence [GK: 3.2-3.3; AF: 3.1-3.2]
• Cauchy estimates and Liouville's theorem [GK: 3.4; AF: 2.6.2]
• zeros of holomorphic functions [GK: 3.6,5.3; AF: 4.4]
• isolated singularities of holomorphic functions [GK: 4.1; AF: 3.5]
• Laurent series and meromorphic functions [GK: 4.2-4.3; AF: 3.3]
4. Residue calculus for meromorphic functions
• Cauchy residue theorem [GK: 4.4-4.5; AF: 4.1]
• evaluation of real definite integrals [GK: 4.6; AF: 4.2]
• evaluation of principal value integrals and semidefinite integrals [GK: 4.6; AF: 4.3]
• singularities at infinity and stereographic projection [GK: 4.7; AF: 1.2.2,4.1]
• argument principle for meromorphic functions [GK: 5.1; AF: 4.4]
5. Harmonic functions and the maximum principle
• properties of harmonic functions [GK: 5.4,7.1, 7.2; AF: 2.6.2]
• the Poisson integral formula [GK: 7.3; AF: 4.6]
• the Dirichlet problem and subharmonic functions [GK: 7.7-7.8]
6. Conformal mappings and linear fractional transformations
• conformal transformations [GK: 6.1; AF: 5.2]
• properties of linear fractional transformations [GK: 6.2-6.3; AF: 5.7]
• critical points and Riemann mapping theorem [GK: 5.2,6.4,7.9; AF: 5.3,5.5]
• Schwarz--Christoffel transformations [GK: 5.5,7.5; AF: 5.6,5.7]
7. Analytic continuation and special functions
• analytic continuation and natural barriers [GK: 10.1-10.3; AF: 3.2,3.5.1]
• multivalued functions, branch cuts, and Riemann surfaces [GK: 10.4; AF: 2.2-2.3]
• elliptic functions and Mittag-Leffler expansions [GK: 10.6; AF: 3.6,5.6]
• properties of the Gamma and Zeta functions [GK: 15.1-15.2; AF: 3.6,4.5]
8. Riemann-Hilbert problems of complex analysis
• Cauchy type integrals and projection operators [AF: 7.1-7.2]
• scalar Riemann-Hilbert problems [AF: 7.3]
• the Hilbert and D-bar problems [AF: 7.4,7.6]