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MTH632 : Complex Analysis and Differential Geometry

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Dr. Sohail Iqbal
Doctor of Philosophy in Mathematics
University of Warwick, Coventry, UK

Course Contents

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.