Cover -- Preface -- Contents -- Chapter 1 Matrix Algebra -- 1.1 Matrix Algebra -- 1.2 Matrix Operations -- 1.3 Properties -- 1.4 Square Matrices -- 1.5 Eigenvalues and Eigenvectors -- Chapter 2 Determinants -- 2.1 Homogeneous Linear Equations -- 2.2 Properties of the Determinant -- 2.3 Applications -- Chapter 3 Vector Derivatives -- 3.1 The Gradient -- 3.2 The Divergence -- 3.3 The Curl -- 3.4 The Product Rules -- 3.5 Derivatives of the Second Order -- 3.6 Applications -- Chapter 4 Gauss, Green and Stokes' Theorem -- 4.1 Line, Surface and Volume Integrals -- 4.2 Gauss' Divergence Theorem -- 4.3 Green's Theorem -- 4.4 Stokes' Curl Theorem (Relation between Line and Surface Integrals) -- Chapter 5 Dirac Delta Function -- 5.1 General Behavior of Delta Function -- 5.2 Generalised Fourier Series -- 5.3 Fourier Transform and Dirac Delta Function -- Chapter 6 Differential Calculus -- 6.1 Operators and Eigenvalues -- 6.2 Expectation Value -- 6.3 Separation of Variables -- 6.4 Wave Function -- 6.5 Application of Differential Equations in Wave Mechanics -- 6.6 Linear Differential Equation with Constant Coefficients -- 6.7 Series Solutions -- Chapter 7 Frobenius Method -- 7.1 The Starting Point -- 7.2 Indicial Equation -- 7.3 Recurrence Relation -- 7.4 Application -- Chapter 8 Convergence -- 8.1 Uniform Convergence -- 8.2 Convergence of a Functional Series -- 8.3 Convergence in the Mean -- 8.4 Convergence Test -- Chapter 9 Orthogonality -- 9.1 The Starting Point -- 9.2 Application -- Chapter 10 Wronskian -- 10.1 Solutions Having Linear Independence -- 10.2 Application -- Chapter 11 Analytic Function -- 11.1 Analyticity and Derivatives of f(z) -- 11.2 Harmonic Functions -- 11.3 Contour Integrals -- 11.4 Integral Theorem of Cauchy -- 11.5 Integral Formula of Cauchy -- Chapter 12 Taylor Series -- 12.1 The Starting Point -- 12.2 Applications.

Chapter 13 Laurent Expansion -- 13.1 The Starting Point -- 13.2 Application -- Chapter 14 Singularity -- 14.1 Some Points About Singularity -- 14.2 Singularity as X -- 14.3 Isolated Singularities -- 14.4 Simple Pole or Pole -- 14.5 Essential Singularity -- 14.6 Branch Point Singularity -- 14.7 Application -- Chapter 15 Calculus of Residues (Cauchy-Riemann) -- 15.1 mth-Order Pole -- 15.2 Simple Pole -- 15.3 Cauchy Residue Theorem -- 15.4 Cauchy's Principal Value -- Chapter 16 Hermite Polynomial -- 16.1 Harmonic Oscillator and Hermite Equation -- 16.2 Solution of Hermite's Equation by a Polynomial Series -- Chapter 17 Legendre Polynomial -- 17.1 The Starting Point -- 17.2 Applications -- Chapter 18 Laguerre Polynomial -- 18.1 The Starting Point -- 18.2 Associated Laguerre Equation -- 18.3 Application -- Chapter 19 Chebyshev Polynomial -- 19.1 The Generating Function -- 19.2 Applications -- Chapter 20 Bessel Function -- 20.1 The Starting Point -- 20.2 Application -- Chapter 21 Fourier Series -- 21.1 Concepts About Fourier Series -- 21.2 The Theorem -- 21.3 Evaluation of The Coefficients -- 21.4 General Form of Fourier Theorem -- 21.5 General Features of Fourier Series -- 21.6 Advantages of Fourier Series -- 21.7 Some Properties of Fourier Series -- 21.8 The Fourier Coefficients Revisited -- Chapter 22 Integral Transform and Kernels -- Chapter 23 Fourier Transform -- 23.1 The Starting Point -- 23.2 Interpreting the Fourier Transform -- 23.3 About the Fourier Integral -- 23.4 Fourier Integral Development -- 23.5 Exponential Form of Fourier Integral -- 23.6 Inverse Fourier Transform -- 23.7 Existence of the Fourier Integral -- 23.8 Properties of Fourier Transform -- 23.9 Derivatives of Fourier Transform -- Chapter 24 Convolution Theorem -- 24.1 Convolution in Fourier Transform -- 24.2 The Causality -- 24.3 Convolution for Laplace Transform.

Chapter 25 Parseval Relation -- 25.1 Parseval Formula for Fourier Series -- 25.2 Parseval Relation for Fourier Transform -- 25.3 Parseval Relation for Hilbert Transform -- Chapter 26 Laplace Transform -- 26.1 The Starting Point -- 26.2 Construction of a Laplace Transform -- 26.3 Inverse Laplace Transform -- 26.4 Laplace Transform of Derivatives -- 26.5 Some Properties of Laplace Transform -- 26.6 Applications -- 26.7 Use of Heaviside's Unit Function -- Chapter 27 Hilbert Transform -- 27.1 Mathematical Preamble -- 27.2 Hilbert Transform -- 27.3 Symmetry Relations -- 27.4 Application -- Chapter 28 Tensor Analysis -- 28.1 The Definition of Tensor -- 28.2 Contravariant and Covariant Tensors -- 28.3 Application -- Bibliography -- Index.