 Syllabus  |   Lectures  |   Downloads  |   FAQ  |   Ask a question  |
Course Co-ordinated by IIT Roorkee
 Coordinators IIT Roorkee IIT Roorkee

Mathematics 3 (Web Course)

COURSE OUTLINE

 Sl no. Topics and Contents No of lectures No of Modules 1 Complex Numbers and Complex Algebra: Geometry of complex numbers, Polar form, Powers and roots of complex numbers. 1 1 2 Complex Functions: Limits of Functions, Continuity, Differentiability, Analytic functions, Cauchy-Riemann Equations, Necessary and Sufficient condition for analyticity, Properties of Analytic Functions, Laplace Equation, Harmonic Functions, Finding Harmonic Conjugate functions 5 1 3 Elementary Analytic Functions: Exponential, Trigonometric, Hyperbolic functions and its properties. Multiple valued function and its branches - Logarithmic function and Complex Exponent function. 4 1 4 Complex Integration: Curves, Line Integrals (contour integral) and its properties. Line integrals of single valued functions, Line integrals of multiple valued functions (by choosing suitable branches). Cauchy-Goursat Theorem, Cauchy Integral Formula, Liouville, FTA, Max/Min Modulus Theorems. 5 1 5 Power Series: Convergence (Ordinary, Uniform, Absoulte) of power series, Taylor and Laurent Theorems, Finding Laurent series expansions. 2 1 6 Zeros, Singularities, Residues: Zeros of analytic functions, Singularities and its properties, Residues, Residue Theorem, Rouche’s Theorem, Argument Principle. 2 1 7 Applications of Contour Integration: Evaluating various type of indefinite real integrals using contour integration method. 4 1 8 Conformal Mapping and its applications: Mappings by elementary functions, Mobius transformations, Schwarz-Christofel transformation, Poisson formula, Dirichlet and Neumann Problems. 5 1 9 Solution in Series: Second order linear equations with ordinary points, Legendre equation, Second order equations with regular singular points, The method of Frobenius, Bessel equation. 4 1 10 Properties of Legendre Polynomials and Bessel Functions 2 1 11 Fourier Series: Orthogonal Family, Fourier Series of 2? periodic functions, Formula for Fourier Coefficients, Fourier series of Odd and Even functions, Half-range series, Fourier series of a T-periodic function, Convergence of Fourier Series, Gibb’s Phenomenon, Differentiation and Integration of Fourier series, Complex form of Fourier series. 4 1 12 Fourier Transforms: Fourier Integral Theorem, Fourier Transforms, Properties of Fourier Transform, Convolution and its physical interpretation, Statement of Fubini’s theorem, Convolution theorems, Inversion theorem, Laplace Transform. 4 1 13 Second order PDE: Second order PDE and classification of 2nd order quasi-linear PDE (canonical form) 1 1 14 Wave Equation: Modeling a vibrating string, D’Alembert’s solution, Duhamel’s principle for one-dimensional wave equation. 2 1 15 Heat Equation: Heat equation, Solution by separation of variables. 2 1 16 Laplace Equation: Laplace Equation in Cartesian, Cylindrical polar and Spherical polar coordinates, Solution by separation of variables. 3 1 17 Solution by Transform Methods: Solutions of PDEs by Fourier and Laplace Transform methods. 2 1

Under development
 Important: Please enable javascript in your browser and download Adobe Flash player to view this site Site Maintained by Web Studio, IIT Madras. Contact Webmaster: nptel@iitm.ac.in