# 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 $${\mathbf{C}}$$

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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