NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (5/9)

The Concept of Analytic Functions

•       Complex numbers, Complex planes, Complex functions

•    Analytic functions

•    Entire functions

•    Harmonic functions

•    Elementary functions: Trigonometric, Complex exponential, Logarithmic and hyperbolic functions

Infinite Series

•    Power series, Derived series, Radius of convergence

•    Taylor series and Laurent series

Conformal Representation

•    Transformation, conformal transformation

•    Linear transformation

•    Möbius transformations

Complex Integration

•    Complex  integrals

•    Cauchy-Goursat theorem

•    Cauchy’s integral formula and their consequences

•    Liouville’s theorem

•    Morera’s theorem

•    Derivative of an analytic function

Singularity and Poles

•    Review of Laurent series

•    Zeros, Singularities

•    Poles and residues

•    Cauchy’s residue theorem

•    Contour Integration

Expansion of Functions and Analytic Continuation

•    Mittag-Leffler theorem

•    Weierstrass’s  factorization theorem

•    Analytic continuation



Section-II (4/9)

Theory of Space Curves

•    Introduction, Index notation and summation convention

•    Space curves, Arc length, Tangent, Normal and binormal

•    Osculating, Normal and rectifying planes

•    Curvature and torsion

•    The Frenet-Serret theorem

•    Natural equation of a curve

•    Involutes and evolutes, Helices

•    Fundamental existence theorem of space curves

Theory of Surfaces

•    Coordinate transformation

•    Tangent plane and surface normal

•    The first fundamental form and the metric tensor

•    The second fundamental form

•    Principal, Gaussian, Mean, Geodesic and normal curvatures

•    Gauss and Weingarten equations

•    Gauss and Codazzi equations