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Mathematical Methods in Science and Engineering, 2nd Edition

Book Description

A Practical, Interdisciplinary Guide to Advanced Mathematical Methods for Scientists and Engineers

Mathematical Methods in Science and Engineering, Second Edition, provides students and scientists with a detailed mathematical reference for advanced analysis and computational methodologies. Making complex tools accessible, this invaluable resource is designed for both the classroom and the practitioners; the modular format allows flexibility of coverage, while the text itself is formatted to provide essential information without detailed study. Highly practical discussion focuses on the “how-to” aspect of each topic presented, yet provides enough theory to reinforce central processes and mechanisms. 

Recent growing interest in interdisciplinary studies has brought scientists together from physics, chemistry, biology, economy, and finance to expand advanced mathematical methods beyond theoretical physics. This book is written with this multi-disciplinary group in mind, emphasizing practical solutions for diverse applications and the development of a new interdisciplinary science.

Revised and expanded for increased utility, this new Second Edition:

  • Includes over 60 new sections and subsections more useful to a multidisciplinary audience
  • Contains new examples, new figures, new problems, and more fluid arguments
  • Presents a detailed discussion on the most frequently encountered special functions in science and engineering
  • Provides a systematic treatment of special functions in terms of the Sturm-Liouville theory
  • Approaches second-order differential equations of physics and engineering from the factorization perspective
  • Includes extensive discussion of coordinate transformations and tensors, complex analysis, fractional calculus, integral transforms, Green's functions, path integrals, and more

Extensively reworked to provide increased utility to a broader audience, this book provides a self-contained three-semester course for curriculum, self-study, or reference. As more scientific disciplines begin to lean more heavily on advanced mathematical analysis, this resource will prove to be an invaluable addition to any bookshelf. 

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
    1. About the Book
    2. About the Second Edition
    3. Acknowledgments
  5. Chapter 1: Legendre Equation and Polynomials
    1. 1.1 Second-Order Differential Equations of Physics
    2. 1.2 Legendre Equation
    3. 1.3 Legendre Polynomials
    4. 1.4 Associated Legendre Equation and Polynomials
    5. 1.5 Spherical Harmonics
    6. Bibliography
    7. Problems
  6. Chapter 2: Laguerre Polynomials
    1. 2.1 Central Force Problems in Quantum Mechanics
    2. 2.2 Laguerre Equation and Polynomials
    3. 2.3 Associated Laguerre Equation and Polynomials
    4. Bibliography
    5. Problems
  7. Chapter 3: Hermite Polynomials
    1. 3.1 Harmonic Oscillator in Quantum Mechanics
    2. 3.2 Hermite Equation and Polynomials
    3. Bibliography
    4. Problems
  8. Chapter 4: Gegenbauer and Chebyshev Polynomials
    1. 4.1 Wave Equation on a Hypersphere
    2. 4.2 Gegenbauer Equation and Polynomials
    3. 4.3 Chebyshev Equation and Polynomials
    4. Bibliography
    5. Problems
  9. Chapter 5: Bessel Functions
    1. 5.1 Bessel's Equation
    2. 5.2 Bessel Functions
    3. 5.3 Modified Bessel Functions
    4. 5.4 Spherical Bessel Functions
    5. 5.5 Properties of Bessel Functions
    6. 5.6 Transformations of Bessel Functions
    7. Bibliography
    8. Problems
  10. Chapter 6: Hypergeometric Functions
    1. 6.1 Hypergeometric Series
    2. 6.2 Hypergeometric Representations of Special Functions
    3. 6.3 Confluent Hypergeometric Equation
    4. 6.4 Pochhammer Symbol and Hypergeometric Functions
    5. 6.5 Reduction of Parameters
    6. Bibliography
    7. Problems
  11. Chapter 7: Sturm–Liouville Theory
    1. 7.1 Self-Adjoint Differential Operators
    2. 7.2 Sturm–Liouville Systems
    3. 7.3 Hermitian Operators
    4. 7.4 Properties of Hermitian Operators
    5. 7.5 Generalized Fourier Series
    6. 7.6 Trigonometric Fourier Series
    7. 7.7 Hermitian Operators in Quantum Mechanics
    8. Bibliography
    9. Problems
  12. Chapter 8: Factorization Method
    1. 8.1 Another Form for the Sturm–Liouville Equation
    2. 8.2 Method of Factorization
    3. 8.3 Theory of Factorization and the Ladder Operators
    4. 8.4 Solutions via the Factorization Method
    5. 8.5 Technique and the Categories of Factorization
    6. 8.6 Associated Legendre Equation (Type A)
    7. 8.7 Schrödinger Equation and Single-Electron Atom (Type F)
    8. 8.8 Gegenbauer Functions (Type A)
    9. 8.9 Symmetric Top (Type A)
    10. 8.10 Bessel Functions (Type C)
    11. 8.11 Harmonic Oscillator (Type D)
    12. 8.12 Differential Equation for the Rotation Matrix
    13. Bibliography
    14. Problems
  13. Chapter 9: Coordinates and Tensors
    1. 9.1 Cartesian Coordinates
    2. 9.2 Orthogonal Transformations
    3. 9.3 Cartesian Tensors
    4. 9.4 Cartesian Tensors and the Theory of Elasticity
    5. 9.5 Generalized Coordinates and General Tensors
    6. 9.6 Operations with General Tensors
    7. 9.7 Curvature
    8. 9.8 Spacetime and Four-Tensors
    9. 9.9 Maxwell's Equations in Minkowski Spacetime
    10. Bibliography
    11. Problems
  14. Chapter 10: Continuous Groups and Representations
    1. 10.1 Definition of a Group
    2. 10.2 Infinitesimal Ring or Lie Algebra
    3. 10.3 Lie Algebra of the Rotation Group
    4. 10.4 Group Invariants
    5. 10.5 Unitary Group in Two Dimensions
    6. 10.6 Lorentz Group and Its Lie Algebra
    7. 10.7 Group Representations
    8. 10.8 Representations of
    9. 10.9 Irreducible Representations of
    10. 10.10 Relation of and
    11. 10.11 Group Spaces
    12. 10.12 Hilbert Space and Quantum Mechanics
    13. 10.13 Continuous Groups and Symmetries
    14. Bibliography
    15. Problems
  15. Chapter 11: Complex Variables and Functions
    1. 11.1 Complex Algebra
    2. 11.2 Complex Functions
    3. 11.3 Complex Derivatives and Cauchy–Riemann Conditions
    4. 11.4 Mappings
    5. Bibliography
    6. Problems
  16. Chapter 12: Complex Integrals and Series
    1. 12.1 Complex Integral Theorems
    2. 12.2 Taylor Series
    3. 12.3 Laurent Series
    4. 12.4 Classification of Singular Points
    5. 12.5 Residue Theorem
    6. 12.6 Analytic Continuation
    7. 12.7 Complex Techniques in Taking Some Definite Integrals
    8. 12.8 Gamma and Beta Functions
    9. 12.9 Cauchy Principal Value Integral
    10. 12.10 Integral Representations of Special Functions
    11. Bibliography
    12. Problems
  17. Chapter 13: Fractional Calculus
    1. 13.1 Unified Expression of Derivatives and Integrals
    2. 13.2 Differintegrals
    3. 13.3 Other Definitions of Differintegrals
    4. 13.4 Properties of Differintegrals
    5. 13.5 Differintegrals of Some Functions
    6. 13.6 Mathematical Techniques with Differintegrals
    7. 13.7 Caputo Derivative
    8. 13.8 Riesz Fractional Integral and Derivative
    9. 13.9 Applications of Differintegrals in Science and Engineering
    10. Bibliography
    11. Problems
  18. Chapter 14: Infinite Series
    1. 14.1 Convergence of Infinite Series
    2. 14.2 Absolute Convergence
    3. 14.3 Convergence Tests
    4. 14.4 Algebra of Series
    5. 14.5 Useful Inequalities About Series
    6. 14.6 Series of Functions
    7. 14.7 Taylor Series
    8. 14.8 Power Series
    9. 14.9 Summation of Infinite Series
    10. 14.10 Asymptotic Series
    11. 14.11 Method of Steepest Descent
    12. 14.12 Saddle-Point Integrals
    13. 14.13 Padé Approximants
    14. 14.14 Divergent Series in Physics
    15. 14.15 Infinite Products
    16. Bibliography
    17. Problems
  19. Chapter 15: Integral Transforms
    1. 15.1 Some Commonly Encountered Integral Transforms
    2. 15.2 Derivation of the Fourier Integral
    3. 15.3 Fourier and Inverse Fourier Transforms
    4. 15.4 Conventions and Properties of the Fourier Transforms
    5. 15.5 Discrete Fourier Transform
    6. 15.6 Fast Fourier Transform
    7. 15.7 Radon Transform
    8. 15.8 Laplace Transforms
    9. 15.9 Inverse Laplace Transforms
    10. 15.10 Laplace Transform of a Derivative
    11. 15.11 Relation Between Laplace and Fourier Transforms
    12. 15.12 Mellin Transforms
    13. Bibliography
    14. Problems
  20. Chapter 16: Variational Analysis
    1. 16.1 Presence of One Dependent and One Independent Variable
    2. 16.2 Presence of More than One Dependent Variable
    3. 16.3 Presence of More than One Independent Variable
    4. 16.4 Presence of Multiple Dependent and Independent Variables
    5. 16.5 Presence of Higher-Order Derivatives
    6. 16.6 Isoperimetric Problems and the Presence of Constraints
    7. 16.7 Applications to Classical Mechanics
    8. 16.8 Eigenvalue Problems and Variational Analysis
    9. 16.9 Rayleigh–Ritz Method
    10. 16.10 Optimum Control Theory
    11. 16.11 Basic Theory: Dynamics versus Controlled Dynamics
    12. Bibliography
    13. Problems
  21. Chapter 17: Integral Equations
    1. 17.1 Classification of Integral Equations
    2. 17.2 Integral and Differential Equations
    3. 17.3 Solution of Integral Equations
    4. 17.4 Hilbert–Schmidt Theory
    5. 17.5 Neumann Series and the Sturm–Liouville Problem
    6. 17.6 Eigenvalue Problem for the Non-Hermitian Kernels
    7. Bibliography
    8. Problems
  22. Chapter 18: Green's Functions
    1. 18.1 Time-Independent Green's Functions in One Dimension
    2. 18.2 Time-Independent Green's Functions in Three Dimensions
    3. 18.3 Time-Independent Perturbation Theory
    4. 18.4 First-Order Time-Dependent Green's Functions
    5. 18.5 Second-Order Time-Dependent Green's Functions
    6. Bibliography
    7. Problems
  23. Chapter 19: Green's Functions and Path Integrals
    1. 19.1 Brownian Motion and the Diffusion Problem
    2. 19.2 Methods of Calculating Path Integrals
    3. 19.3 Path Integral Formulation of Quantum Mechanics
    4. 19.4 Path Integrals Over Lévy Paths and Anomalous Diffusion
    5. 19.5 Fox's -Functions
    6. 19.6 Applications of -Functions
    7. 19.7 Space Fractional Schrödinger Equation
    8. 19.8 Time Fractional Schrödinger Equation
    9. Bibliography
    10. Problems
  24. Further Reading
  25. Index
  26. End User License Agreement