Calculus and Undergraduate Analysis
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Continuity and differentiation in one and several real variables
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Inverse and implicit function theorems
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Compactness and connectedness in analysis
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Uniform convergence and uniform continuity
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Riemann integrals
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Contour integrals and Green’s theorem
Reference: (3).
Preliminary Topics in Complex Analysis
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Complex arithmetic
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Analyticity, harmonic functions, and the Cauchy-Riemann equations
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Contour Integration in \({\mathbf{C}}\)
References: (1) Chapters 1, 2; (2) Chapters 1, 2, 4; (4) Chapter 1.
Cauchy’s Theorem and its consequences
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Cauchy’s theorem and integral formula,
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Morera’s theorem,
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Schwarz reflection
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Uniform convergence of analytic functions
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Taylor and Laurent expansions
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Maximum modulus principle and Schwarz’s lemma
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Liouville’s theorem and the Fundamental theorem of algebra
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Residue theorem and applications
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Singularities and meromorphic functions
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Casorati-Weierstrass theorem
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Rouche’s theorem, the argument principle, and the open mapping theorem
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Estimates using Cauchy Integral Formula:
- Cauchy inequalities
- More generally, bounds on holomorphic functions and their derivatives on compact sets
References: (1) Chapters 4, 5, 6; (2) Chapters 5, 7, 8, 9; (4) Chapters 2, 3, 5, 8 (§2,3).
Conformal Mapping
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General properties of conformal mappings
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Analytic and mapping properties of linear fractional transformations
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Automorphisms of the disk, plane, and Riemann sphere
References: (1) Chapters 3, 8; (2) Chapters 3, 4; (4) Chapter 8 (§1,2).
References
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L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill.
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E. Hille, Analytic Function Theory, Vol. 1, Ginn and Company.
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W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill.
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E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.