No Chalk Academy

Complex Analysis

Good Books for reference:

Fundamentals of complex analysis (E.B. Saff and A. D. Snider)
Complex variables and applications (Churchill and Brown)
Functions of one complex variable (J. B. Conway)

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Syllabus for complex analysis

Analytic functions

Harmonic functions

Lecture 1: Complex Analysis (History of Complex Numbers.)

Lecture 2: Complex Analysis (How complex numbers appeared while solving algebraic equations?)

Lecture 3: Complex Analysis (Set of complex numbers.)

Lecture 4: Complex Analysis (Algebra of complex numbers.)

Lecture 5: Complex Analysis (Polar form of complex numbers.)

Lecture 6: Complex Analysis (Polar form in terms of Exponential function.)

Lecture 7: Complex Analysis (nth power and mth root of complex numbers using their polar forms.)

Lecture 8: Complex Analysis (Definition of DOMAINS.)

Lecture 9: Complex Analysis (Importance of DOMAINS.)

Lecture 10: Complex Analysis (Stereographic projection.)

Lecture 11: Complex Analysis (Complex valued functions of complex variables I.)

Lecture 12: Complex Analysis (Complex valued functions of complex variables II.)

Lecture 13: Complex Analysis (Complex valued functions of complex variables III.)

Lecture 14:Complex Analysis (Defining limits of complex functions.)

Lecture 15: Complex Analysis (Continuity of complex functions.)

Lecture 16: Complex Analysis (admissible and inadmissible functions.)

Lecture 17: Complex Analysis (differentiable/ analytic functions.)

Lecture 18: Complex Analysis (Cauchy Riemann equations)

Lecture 19: Complex Analysis (CR conditions are not sufficient)

Lecture 20: Complex Analysis (When do the CR conditions become sufficient?)

Lecture 21: Complex Analysis(Derivative of a function zero in a domain implies it is constant)

Lecture 22: Complex Analysis(Analytic to harmonic functions and vice versa)

Lecture 23: Complex Analysis (Level curves of real and imaginary parts of analytic functions)

Lecture 24: Complex Analysis (Polynomial functions.)

Lecture 25: Complex Analysis (Rational functions)

Lecture 26: Complex Analysis (Exponential function.)

Lecture 27: Complex Analysis (Trigonometric and hyperbolic functions.)

Complex integration

Cauchy’s integral theorem and formula

Liouville’s theorem

Maximum modulus principle

Morera’s theorem

Zeros and singularities

Summary lecture: Complex Integration 1

Summary lecture: Complex Integration 2

Summary lecture: Complex Integration 3

Summary lecture: Complex Integration 4

Summary lecture:Complex integration 5 (CAUCHY INTEGRAL FORMULA FOR DERIVATIVES)

Summary lecture:Complex integration 6 (CAUCHY INTEGRAL FORMULA TO COMPUTE REAL INTEGRALS)

Summary lecture:Complex integration 7 (CONSEQUENCES OF CAUCHY'S INTEGRAL FORMULA)

Summary lecture:Complex integration 8 (MAXIMUM MODULUS PRINCIPLE)

Power series

Radius of convergence

Taylor’s theorem and Laurent’s theorem

Residue theorem and applications for evaluating real integrals

Rouche’s theorem

Argument principle

Schwarz lemma

Conformal mappings

Bilinear transformations

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