Complex Numbers and Complex Algebra:
Geometry of complex numbers, Polar form, Powers and roots of complex numbers.

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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

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Elementary Analytic Functions:
Exponential, Trigonometric, Hyperbolic functions and its properties. Multiple valued function and its branches - Logarithmic function and Complex Exponent function.

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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.

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Power Series:
Convergence (Ordinary, Uniform, Absoulte) of power series, Taylor and Laurent Theorems, Finding Laurent series expansions.

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Zeros, Singularities, Residues:
Zeros of analytic functions, Singularities and its properties, Residues, Residue Theorem, Rouche’s Theorem, Argument Principle.

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Applications of Contour Integration:
Evaluating various type of indefinite real integrals using contour integration method.

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Conformal Mapping and its applications:
Mappings by elementary functions, Mobius transformations, Schwarz-Christofel transformation, Poisson formula, Dirichlet and Neumann Problems.

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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.

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Properties of Legendre Polynomials and Bessel Functions

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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.

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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.

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Second order PDE:
Second order PDE and classification of 2nd order quasi-linear PDE (canonical form)

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Wave Equation:
Modeling a vibrating string, D’Alembert’s solution, Duhamel’s principle for one-dimensional wave equation.

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Heat Equation:
Heat equation, Solution by separation of variables.

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Laplace Equation:
Laplace Equation in Cartesian, Cylindrical polar and Spherical polar coordinates, Solution by separation of variables.

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Solution by Transform Methods:
Solutions of PDEs by Fourier and Laplace Transform methods.

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