Syllabus  |   Lectures  |   Downloads  |   FAQ  |   Ask a question  |  
Course Co-ordinated by IIT Roorkee
Coordinators
 
Prof. P.N. Agrawal
IIT Roorkee

 
Dr. Tanuja Srivastava
IIT Roorkee

 

Download Syllabus in PDF format



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