Complex Analysis

Calculus and Undergraduate Analysis

  • Continuity and differentiation in one and several real variables

  • Inverse and implicit function theorems

  • Compactness and connectedness in analysis

  • Uniform convergence and uniform continuity

  • Riemann integrals

  • Contour integrals and Green’s theorem

Reference: (3).

Preliminary Topics in Complex Analysis

  • Complex arithmetic

  • Analyticity, harmonic functions, and the Cauchy-Riemann equations

  • Contour Integration in \({\mathbb{C}}\)

References: (1) Chapters 1, 2; (2) Chapters 1, 2, 4; (4) Chapter 1.

Cauchy’s Theorem and its consequences

  • Cauchy’s theorem and integral formula,

  • Morera’s theorem,

  • Schwarz reflection

  • Uniform convergence of analytic functions

  • Taylor and Laurent expansions

  • Maximum modulus principle and Schwarz’s lemma

  • Liouville’s theorem and the Fundamental theorem of algebra

  • Residue theorem and applications

  • Singularities and meromorphic functions

  • Casorati-Weierstrass theorem

  • Rouche’s theorem, the argument principle, and the open mapping theorem

  • 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

  • General properties of conformal mappings

  • Analytic and mapping properties of linear fractional transformations

  • Automorphisms of the disk, plane, and Riemann sphere

References: (1) Chapters 3, 8; (2) Chapters 3, 4; (4) Chapter 8 (§1,2).

References

  • L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill.

  • E. Hille, Analytic Function Theory, Vol. 1, Ginn and Company.

  • W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill.

  • E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.

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