Complex Analysis II
Semester IV (PS04CMTH51)
Syllabus
| Unit-I | Riemann Stieltjes integral: a function of bounded variation on \([a,b]\), its total variation, rectifiable curve, smooth curve, piecewise smooth curve, polygonal path, integral of a continuous function on \([a,b]\) with respect to a function of bounded variation and its properties, integral of continuous function defined on \(\{\gamma\}\) with respect to \(\gamma\) and its properties, zeros of an analytic function, multiplicity of zero of an analytic function, the index of a closed curve and its properties. |
| Unit-II | Cauchy’s Theorem (First version), Cauchy’s Integral Formula (First and Second Version), Cauchy’s Integral formula for derivatives, Morera’s Theorem, existence of a primitive on simply connected region, characterization of non-vanishing analytic function on simply connected region, Counting zero principle and open mapping theorem, Classification of singularities namely removable singularity, pole and essential singularity, order of a pole, Casorati-Weierstrass theorem. |
| Unit-III | Argument Principle, its generalization and examples, Rouche’s theorem and deduction of Fundamental Theorem of Algebra, Maximum Modulus principle (statements only), Schwarz’s lemma, its applications and consequences, Mobius transformation \(\varphi_a\) and its properties, the space of continuous functions \(C(G,\Omega)\), the topology on \(C(G,\Omega)\), normal family in \(C(G,\Omega)\), equicontinuity of a family in \(C(G,\Omega)\), Arzela Ascoli theorem in \(C(G,\Omega)\). |
| Unit-IV | The space \(H(G)\) of analytic functions, locally bounded family in \(H(G)\), Hurwitz’s therorem, Montel’s theorem, infinite product, convergence and absolute convergence of infinite product, convergence of infinite product of elements in \(H(G)\), elementary factors and its properties, The Weierstrass Factorization Theorem, factorization of \(\sin\), \(\cos\), \(\sinh\) and \(\cosh\), Walli’s formula. |
Reference Books
- J. B. Conway, Functions of One Complex Variable, 2nd Edition, Narosa, New Delhi, 1978.
- W. Rudin, Real and Complex Analysis, McGraw Hill, 1967.
- R. Narasimhan and Y. Nievergelt, Complex Analysis in One Variable, Birkhauser, 2001.