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Maths 3
Posted Date: 19 Jan 2008 Resource Type: Articles/Knowledge Sharing Category: Education
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Posted By: Moncy Member Level: Silver
Rating:  Points: 2
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Sl no. Topics and Contents Number of lectures Number 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:
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
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