Analytic functions: Limits and continuity, differentiability and analyticity, analytic branches of inverse of functions, branches of logarithm, Cauchy-Riemann equations, harmonic conjugates. Complex integral: Cauchy’s theorem and integral formula, series of complex functions and the Weierstrass M-test, Taylor series, identity theorem, isolation of zeros of an analytic function, statements of open mapping, inverse function, Liouville’s theorem, fundamental theorem of Algebra.
Residue Calculus: Singularities and their classification, Laurent series, residue theorem and argument principle, computing real integrals using residues. Bilinear transformation: Bilinear transformation, conformal mapping, elementary properties of the mapping of exponential, sine and cosine functions.
- E. Kreyszig, Advanced engineering mathematics, 10th Edition, John Wiley & sons (2011)
- R.V Churchill and J.W. Brown: Complex Variables and Applications, 8th Edition, Mc-Graw Hill (2009).
- S. Ponnusamy and H. Silverman, Complex Variables with Applications, Birkhauser (2006)