Mathematics in Physics and Engineering

Mathematics in Physics and Engineering describes the analytical and numerical (desk-machine) methods that arise in pure and applied science, including wave equations, Bessel and Legendre functions, and matrices. The manuscript first discusses partial differential equations, as well as the method of separation of variables, three-dimensional wave equation, diffusion or heat flow equation, and wave equation in plane and cylindrical polar coordinates. The text also ponders on Frobenius' and other methods of solution. Discussions focus on hypergeometric equation, Bessel's equation, confluent hypergeometric equation, and change of dependent and independent variables. The publication takes a look at Bessel and Legendre functions and Laplace and other transforms, including orthogonal properties, applications from electromagnetism, spherical harmonics, and application to partial differential equations. The book also examines matrices, analytical methods in classical and wave mechanics, calculus of variations, and complex variable theory and conformal transformations. The book is a dependable reference for mathematicians, engineers, and physicists both at undergraduate and postgraduate levels.

Contents Preface Chapter I. Introduction To Partial Differential Equations 1. Introduction 2. The One-Dimensional Wave Equation 3. Method Of Separation Of Variables 4. The Two-Dimensional Wave Equation 5. Three-Dimensional Wave Equation 6. The Wave Equation In Plane And Cylindrical Polar Coordinates A. Plane Polars B. Cylindrical Polars 7. The Wave Equation In Spherical Polar Coordinates 8. Laplace's Equation In Two Dimensions A. Cartesian Coordinates B. Polar Coordinates 9. Laplace's Equation In Three Dimensions 10. The Diffusion Or Heat Flow Equation 10.1. Neutron Diffusion 11. A Fourth Order Partial Differential Equation 12. The Bending Of An Elastic Plate — The Biharmonic Equation 13. Characteristics 13.1. Cauchy's Problem 13.2. Reduction Of (13.1.1) To The Standard Form 13.3. Riemann's Method Of Solution Of (13.1.1) 13.4. Numerical Integration Of Hyperbolic Differential Equations Problems General References Chapter II. Ordinary Differential Equations: Frobenius' And Other Methods Of Solution 1. Introduction 2. Solution In Series By The Method Of Frobenius 3. Bessel's Equation 4. Legendre's Equation 5. Hyper Geometric Equation 6. Series Solution About A Point Other Than The Origin 6.1. The Transformation X = (1 - ξ)/2 7. Series Solution In Descending Powers Of X 8. Confluent Hypergeometric Equation 8.1. Laguerre Polynomials 8.2. Hermite Polynomials 9. Asymptotic Or Semi-Convergent Series 10. Change Of Dependent Variable 11. Change Of The Independent Variable 12. Exact Equations 13. The Inhomogeneous Linear Equation 14. Perturbation Theory For Non-Linear Differential Equations 14.1. The Perturbation Method 14.2. Periodic Solutions Problems General References Chapter III. Bessel And Legendre Functions 1. Definition Of Special Functions 127 2. Jn(X), The Bessel Function Of The First Kind Of Order N 2.1. Recurrence Relations: Jn(X) 3. Bessel Function Of The Second Kind Of Order N, Yn(X) 4. Equations Reducible To Bessel's Equation 5. Applications 6. Modified Bessel Functions: In(X), Kn(X) 6.1. Recurrence Relations For In(X) And Kn(X) 6.2. Equations Reducible To Bessel's Modified Equation 6.3. Bessel Functions Of The Third Kind (Hankel Functions) 7. Illustrations Involving Modified Bessel Functions 8. Orthogonal Properties 8.1. Expansion Of F(X) In Terms Of Jn(ξix) 8.2. Jn(X) As An Integral (Where N Is Zero Or An Integer) 8.3. Other Important Integrals 9. Integrals Involving The Modified Bessel Functions 10. Zeros Of The Bessel Functions 11. A Generating Function For The Legendre Polynomials 11.1. Recurrence Relations 11.2. Orthogonality Relations For The Legendre Polynomials 11.3. Associated Legendre Functions 12. Applications From Electromagnetism 13. Spherical Harmonics 14. The Addition Theorem For Spherical Harmonics Problems General References Chapter IV. The Laplace And Other Transforms 1. Introduction 2. Laplace Transforms And Some General Properties 3. Solution Of Linear Differential Equations With Constant Coefficients 4. Further Theorems And Their Application 5. Solution Of The Equation Φ(D)x(t) = F(t) By Means Of The Convolution Theorem 6. Application To Partial Differential Equations 7. The Finite Sine Transform 8. The Simply Supported Rectangular Plate 9. Free Oscillations Of A Rectangular Plate 10. Plate Subject To Combined Lateral Load And A Uniform Compression 11. The Fourier Transform Problems Chapter V. Matrices 1. Introduction 1.1. Definitions 2. Determinants 2.1. Evaluation Of Determinants 3. Reciprocal Of A Square Matrix 3.1. Determinant Of The Adjoint Matrix 4. Solution Of Simultaneous Linear Equations 4.1. Choleski-Turing Method 4.2. A Special Case: The Matrix A Is Symmetric 5. Eigenvalues (Latent Roots) 5.1. The Cayley-Hamilton Theorem 5.2. Iterative Method For Determination Of Eigenvalues 5.3. Evaluation Of Subdominant Eigenvalue 6. Special Types Of Matrices 6.1. Orthogonal Matrix 6.2. Hermitian Matrix 7. Simultaneous Diagonalization Of Two Symmetric Matrices Problems General References Chapter VI. Analytical Methods In Classical And Wave Mechanics 1. Introduction 2. Definitions 3. Lagrange's Equations Of Motion For Holonomic Systems 3.1. Derivation Of The Equations 3.2. Conservative Forces 3.3. Illustrative Examples 3.4. Energy Equation 3.5. Orbital Motion 3.6. The Symmetrical Top 3.7. The Two-Body Problem 3.8. Velocity-Dependent Potentials 3.9. The Relativistic Lagrangian 4. Hamilton's Equations Of Motion 5. Motion Of A Charged Particle In An Electromagnetic Field 6. The Solution Of The Schrödinger Equation 6.1. The Linear Harmonic Oscillator 6.2. Spherically Symmetric Potentials In Three Dimensions 6.3. Two-Body Problems Problems General References Chapter VII. Calculus Of Variations 1. Introduction 2. The Fundamental Problem: Fixed End-Points 2.1. Special Cases 2.2. Variable End-Points 2.3. A Generalization Of The Fixed End-Point Problem 2.4. One Independent, Several Dependent Variables 2.5. One Dependent And Several Independent Variables 3. Isoperimetric Problems 4. Rayleigh-Ritz Method 4.1. Sturm-Liouville Theory For Fourth-Order Equations 5. Torsion And Viscous Flow Problems 5.1. Torsional Rigidity 5.2. Trefftz Method 5.3. Generalization To Three Dimensions 6. Variational Approach To Elastic Plate Problems 6.1. Boundary Conditions 6.2. Buckling Of Plates 7. Binding Energy Of The He4 Nucleus 8. The Approximate Solution Of Differential Equations Problems General References Chapter VIII. Complex Variable Theory And Conformal Transformations 1. The Argand Diagram 2. Definitions Of Fundamental Operations 3. Function Of A Complex Variable 3.1. Cauchy-Riemann Equations 4. Geometry Of Complex Plane 5. Complex Potential 5.1. Uniform Stream 5.2. Source, Sink And Vortex 5.3. Doublet (Dipole) 5.4. Uniform Flow + Doublet -F Vortex. Flow Past A Cylinder 5.5. A Torsion Problem In Elasticity 6. Conformal Transformation 6.1. Bilinear (Möbius) Transformation 7. Schwarz-Christoffel Transformation 7.1. Applications 7.2. The Kirchhoff Plane 8. Transformation Of A Circle Into An Aerofoil Problems General References Chapter IX. The Calculus Of Residues 1. Definition Of Integration 2. Cauchy's Theorem 3. Cauchy's Integral 3.1. Differentiation 4. Series Expansions 4.1. Laurent's Theorem 5. Zeros And Singularities 5.1. Residues 6. Cauchy Residue Theorem 6.1. Application Of Cauchy's Theorem 6.2. Flow Round A Cylinder 6.3. Definite Integrals. Integration Round Unit Circle 6.4. Infinite Integrals 6.5. Jordan's Lemma 6.6. Another Type Of Infinite Integral 7. Harnack's Theorem And Applications 7.1. The Schwarz And Poisson Formulas 7.2. Application Of Conformai Transformation To Solution Of A Torsion Problem 8. Location Of Zeros Of f(z) 8.1. Nyquist Stability Criterion 9. Summation Of Series By Contour Integration 10. Representation Of Functions By Contour Integrals 10.1. Gamma Function 10.2. Bessel Functions 10.3. Legendre's Function As A Contour Integral 11. Asymptotic Expansions 12. Saddle-Point Method Problems General References Chapter X. Transform Theory 1. Introduction 1.1. Complex Fourier Transform 1.2. Laplace Transform 1.3. Hilbert Transform 1.4. Hankel Transform 1.5. Mellin Transform. 2. Fourier's Integral Theorem 3. Inversion Formulas 3.1. Complex Fourier Transform 3.2. Fourier Sine And Cosine Transforms 3.3. Convolution Theorems For Fourier Transforms 4. Laplace Transform 4.1. The Inversion Integral On The Infinite Circle 4.2. Exercises In The Use Of The Laplace Transform 4.3. Linear Approximation To Axially Symmetrical Supersonic Flow 4.4. Supersonic Flow Round A Slender Body Of Revolution 5. Mixed Transforms 5.1. Linearized Supersonic Flow Past Rectangular Symmetrical Aerofoil 5.2. Heat Conduction In A Wedge 6. Integral Equations 6.1. The Solution Of A Certain Type Of Integral Equation Of The First Kind 6.2. Poisson's Integral Equation 6.3. Abel's Integral Equation 7. Hilbert Transforms 7.1. Infinite Hilbert Transform 7.2. Finite Hilbert Transform 7.3. Alternative Forms Of The Finite Hilbert Transform Problems General References Chapter XI. Numerical Methods 1. Introduction 1.1. Finite Difference Operators 2. Interpolation And Extrapolation 2.1. Linear Interpolation 2.2. Everett's And Bessel's Interpolation Formulas 2.3. Inverse Interpolation 2.4. Lagrange Interpolation Formula 2.5. Formulas Involving Forward Or Backward Differences 3. Some Basic Expansions 4. Numerical Differentiation 5. Numerical Evaluation Of Integrals 5.1. Note On Limits Of Integration 5.2. Evaluation Of Double Integrals 6. Euler-Maclaurin Integration Formula 6.1. Summation Of Series 7. Solution Of Ordinary Differential Equations By Means Of Taylor Series 8. Step-By-Step Method Of Integration For First-Order Equations 8.1. Simultaneous First-Order Equations And Second-Order Equations With The First Derivative Present 8.2. The Second-Order Equation y" = f(x,y) 8.3. Alternative Method For The Linear Equation y" = g(x)y + h(x) 9. Boundary Value Problems For Ordinary Differential Equations Of The Second Order 9.1. Approximate Solution Of Eigenvalue Problems By Finite Differences 9.2. Numerical Solution Of Eigenvalue Equations 10. Linear Difference Equations With Constant Coefficients 11. Finite Differences In Two Dimensions Problems General References Chapter XII. Integral Equations 1. Introduction 1.1. Types Of Integral Equations 1.2. Some Simple Examples Of Linear Integral Equations 2. Volterra Integral Equation Form For A Differential Equation 2.1. Higher Order Equations 3. Fredholm Integral Equation Form For Sturm-Liouville Differential Equations 3.1. The Modified Green's Function 3.2. Green's Function For Fourth-Order Differential Equations 4. Numerical Solution 4.1. The Numerical Solution Of The Homogeneous Equation 4.2. The Volterra Equation 4.3. Iteration Method Of Solution 5. The Variation-Iteration Method For Eigenvalue Problems Problems General References Appendix 1. Δ2Φ In Spherical And Cylindrical Polar Coordinates 1.1. Plane Polar Coordinates 1.2. Cylindrical Polar Coordinates 1.3. Spherical Polar Coordinates 2. Partial Fractions 3. Sequences, Series, And Products 3.1. Sequences 3.2. Series 3.3. Infinite Products 4. Maxima And Minima For Functions Of Two Variables 4.1. Euler's Theorem Of Homogeneous Functions 4.2. The Expansion Of (Sinh aU/)/(Sinh U) In Powers Of 2 Sinh (½) 5. Integration 5.1. Uniform Convergence Of Infinite Integrals 5.2. Change Of Variables In A Double Integral 5.3. Special Integrals 5.4. Elliptic Integrals 6. Principal Valued Integrals 7. Vector Algebra And Calculus 7.1. Curvilinear Coordinates 7.2. The Equation Of Heat Conduction 7.3. Components Of Velocity And Acceleration In Plane Polar Coordinates 7.4. Vectors, Dyads And Tensors 8. Legendre Functions Of Non-Integral Order 8.1. The Value Of Pv(0) 9. An Equivalent Form For F(a,b;c;x) 10. Integrals Involving Ln(k)(x) Problems General References Solutions Of Problems Chapter I Chapter II Chapter III Chapter IV Chapter V Chapter VI Chapter VII Chapter VIII Chapter IX Chapter X Chapter XI Chapter XII Appendix Subject Index

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