Question Bank CVPDE MAT3003.pdf - Question Bank Complex...

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Complex Variables and Partial Differential Equations (MAT3003) Question Bank
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Question Bank
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Contents 1 Analytic functions 1 1.1 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Harmonic Conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Analytic Functions in Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Conformal Mapping 7 2.1 Geometrical Representation of Complex Functions . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 The Squared Mapping w = z 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 The Exponential Function e z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Bilinear Transformations 13 3.1 Linear Fractional Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Finding a Blinear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Images under a Bilinear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Complex Integration 17 4.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Contour integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Cauchy’s Integral Formula 19 5.1 Simply and Multiply Connected Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Taylor’s Series, Laurent’s Series and Residues 23 6.1 Taylor’s Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 Taylor’s Series for Rational Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4 Laurent Series about an Isolated Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.5 Laurent Series about a Point of Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 iii
iv CONTENTS 6.6 Types of Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.7 Formula for the Residue at a Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7 Contour Integration through Cauchy’s Residue Theorem 31 7.1 Cauchy’s Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8 Real Integrals through Residue Theorem 35 8.1 Cauchy’s Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.2 Type 1 - Integrals around the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.3 Type 2 - Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8.4 Type 3 - Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8.5 Type 4 - Using Indented Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 9 Partial Differential Equations and their Formation 47 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 9.2 Formation of a Partial Differential Equation - Elimination of Arbitrary Constants . . . . . . . 48 9.3 Formation of a Partial Differential Equation - Elimination of Arbitrary Functions . . . . . . . 50 10 Partial Differential Equations of First Order 53 10.1 Solution of a Partial Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 10.2 Quasi-linear Partial Differential Equations of First Order . . . . . . . . . . . . . . . . . . . . 53 10.3 Solution of Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 10.4 Nonlinear Partial Differential Equations of First Order . . . . . . . . . . . . . . . . . . . . . . 57 10.5 Special forms of Nonlinear First Order Partial Differential Equations . . . . . . . . . . . . . . 58 11 Linear Partial Differential Equations of Higher Order 67 11.1 Linear Equations of Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 11.2 Linear Equations of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 11.3 Finding the Complementary Function z c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 11.4 Finding the Particular Integral z p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12 Applications of Partial Differential Equations 73 12.1 Vibrating Strings - One dimensional Wave Equation . . . . . . . . . . . . . . . . . . . . . . . 73 12.2 Heat-Flow with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 80 12.3 Heat-Flow with nonhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . 88
CONTENTS v 12.4 Two Dimensional Steady State Heat Flow in Metal Plates . . . . . . . . . . . . . . . . . . . . 89 13 Complex Fourier Transform 95 13.1 Complex Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13.2 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13.3 Fourier Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 14 Fourier Sine and Cosine Transforms 103 14.1 Fourier Sine and Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 15 Parseval’s Identities 107 15.1 Parseval’s Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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Semester: Winter Semester 2019-2020 Course Title: Complex Variables and Partial Differential Equations Course Code: MAT3003 - Theory Syllabus and Organization of Contents Module 1 Analytic Functions Complex variables - Analytic functions and Cauchy-Riemann equations - Laplace equation and harmonic

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