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Syllabus Complex Analysis

Syllabus:

  1. Analytic functions [10 Lectures]
    • Functions of Complex Variables, Limits, theorems on limits, Limits involving the point at infinity, continuity, derivatives, differentiation formulas, Cauchy-Riemann Equations, Sufficient conditions for differentiability, polar coordinates,Harmonic functions.
  2. Elementary Functions [8 Lectures]
    • Elementary Functions, Exponential functions, Logarithmic function and its branches and derivatives of logarithms, some identities involving logarithms, complex exponents, Trigonometric functions, Hyperbolic functions, inverse trigonometric and hyperbolic functions.
  3. Definite Integrals [12 Lectures]
    • Derivatives of functions, definite integrals of functions, contours, contour integrals, examples, upper bounds for moduli of contour integrals, anti-derivatives,examples, Cauchy-Goursat Theorem, Simply and multiply connected domains.Cauchy integral formula. Derivatives of analytic functions. Liouville's Theorem. Fundamental Theorem of Algebra.
  4. Series [8 Lectures]
    • Convergence of sequences, convergence of series, Taylor Series, examples, Laurent Series, examples. Absolute and uniform convergence of power series, continuityof the sum of a power series, Integration and Differentiation of power series.
  5. Residues, Cauchy residue theorem using a single residue, three types of isolated singular points, residues at poles, zeros of analytic functions, zeros  and poles. [6 Lectures]
  6. Applications of Residues [4 Lectures]
    • Evaluation of improper integrals, examples.

Text Book:

  • R.V. Churchill and I.W. Brown, Complex Variables and Applications, International Student Edition, 2003. (Seventh Edition).
    • Chapter 2 : Sections 18 to 25, Chapter 3 : Section 28 to 34, Chapter 4 : Sections 36 to 44 and Sections 46 to 50; (except 45), Chapter 5 : Section 53 to 56, Chapter 6 : Sections 62 to 69, Chapter 7 : Section 72, Chapter 8 : Section 84 to 87.

Reference Books:

  1. S. Ponnusamy, Complex Analysis , Second Edition (Narosa).
  2. J.M. Howie, Complex Analysis, (Springer, 2003).
  3. S. Lang, Complex Analysis, (Springer Verlag).
  4. A.R. Shastri, An Introduction to Complex Analysis, (MacMillan).