**Complex Analysis:**

Representation
of complex numbers, Algebraic properties
of complex numbers, Geometric representation of complex numbers, Complex
conjugates, Exponential forms, Algebraic properties of exponential form, DeMoiver’s Theorem, Regions on complex
plane, Functions of a complex variable, Mappings, Limits, Theorems on limits, Limits
involving the point at infinity, Continuity, Derivative, Differentiation
formulas, Cauchy-Reimann Equations, Cauchy-Reimann Equations in Polar
coordinates, Analytic functions, Harmonic functions, Uniquely determined
analytic functions, Reflection Principle, Exponential function, Logarithmic
functions, Branches and derivatives of Logarithms, Identities involving
Logarithms, Complex exponents, Trigonometric functions, Hyperbolic functions, Inverse
trigonometric and hyperbolic functions, Inverse trigonometric and hyperbolic
functions, Derivative of functions, Definite integrals of functions, Contours, Contour
integrals, Upper bounds for moduli of contour integrals, Antiderivatives, Cauchy-Goursat
theorem, Domains, Cauchy Integral Formula, Liouville’s theorem, Sequence of
complex numbers, Convergence of sequence, Series, Taylor Series, Laurent Series,
Continuity and power series, Differentiation of power series, Integration of
power series, Uniqueness of series representation, Algebra of power series, Isolated
singular points, Residues, Cauchy Residue theorem, Residues at infinity, Isolated
singular points, Residues at poles, Zeros of analytic function, Zeros and poles,
Functions and isolated singular points.

**Differential Geometry:**

Introduction
to differential Geometry, Euclidean space, Space curves, Arc-length, Arc-length Parameterization, Tangent, normal and binormal, Osculating,
Rectifying and normal planes, Curvature, Frenet-Seret Theorem, Torsion, Frenet-Seret
apparatus, Surfaces, Tangent planes, Surface normal, First fundamental form, Second
fundamental form, Shape operators, Normal curvature, Gaussian curvature, Geodesics.