COMPLEX ANALYSIS (Optional Paper)
Paper Code:
MAT327
Credits:
5
Contact Hours:
75.00
Max. Marks:
100.00
Unit I:
I
15.00
Functions of a complex Variable, Differentiability and analyticity, Cauchy Riemann Equations, Harmonic functions, Existence of Harmonic conjugate, transcendental functions such as exponential, trigonometric and hyperbolic functions.
Unit II:
II
15.00
Power series as an analytic function. Properties of line integrals, Goursat Theorem, Cauchy theorem, Consequence of simply connectivity, Index of a closed curve, Cauchy’s integral formula.
Unit III:
III
15.00
Morera’s theorem, Liouville’s theorem, Fundamental theorem of Algebra, Taylor’s theorem, Laurent series, singularities, Classification of singularities, Zeros of Analytic functions.
Unit IV:
IV
15.00
Maximum modulus theorem, Minimum modulus theorem, Hadamard three circle theorem, Schwarz’s Lemma, Rouche’s theorem, Calculation of residues, Residue theorem.
Unit V:
V
15.00
Evaluation of integrals of the form ∫_α^(2π+α)▒〖R(cos θ,sinθ)dθ〗, ∫_(-∞)^∞▒〖f(x)dx〗, Conformal mappings and Mobius transformations.
Essential Readings:
- Complex Analysis (Third edition) by L. V. Ahlfors, McGraw Hill Book Company, 1979.
- Complex Analysis by J. B. Conway, Narosa Publishing House.
- Complex Analysis by Serg Lang, Addison Wesley.
- Foundations of Complex analysis (Second Edition), S. Ponnusamy, Narosa Publishing House.
- Complex variables and Applications by Ruel V. Churchill.
Academic Year:
