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
- Cover
- Title Page
- Copyright
- Preface
- Chapter 1: Legendre Equation and Polynomials
- Chapter 2: Laguerre Polynomials
- Chapter 3: Hermite Polynomials
- Chapter 4: Gegenbauer and Chebyshev Polynomials
- Chapter 5: Bessel Functions
- Chapter 6: Hypergeometric Functions
- Chapter 7: Sturm–Liouville Theory
-
Chapter 8: Factorization Method
- 8.1 Another Form for the Sturm–Liouville Equation
- 8.2 Method of Factorization
- 8.3 Theory of Factorization and the Ladder Operators
- 8.4 Solutions via the Factorization Method
- 8.5 Technique and the Categories of Factorization
- 8.6 Associated Legendre Equation (Type A)
- 8.7 Schrödinger Equation and Single-Electron Atom (Type F)
- 8.8 Gegenbauer Functions (Type A)
- 8.9 Symmetric Top (Type A)
- 8.10 Bessel Functions (Type C)
- 8.11 Harmonic Oscillator (Type D)
- 8.12 Differential Equation for the Rotation Matrix
- Bibliography
- Problems
-
Chapter 9: Coordinates and Tensors
- 9.1 Cartesian Coordinates
- 9.2 Orthogonal Transformations
- 9.3 Cartesian Tensors
- 9.4 Cartesian Tensors and the Theory of Elasticity
- 9.5 Generalized Coordinates and General Tensors
- 9.6 Operations with General Tensors
- 9.7 Curvature
- 9.8 Spacetime and Four-Tensors
- 9.9 Maxwell's Equations in Minkowski Spacetime
- Bibliography
- Problems
-
Chapter 10: Continuous Groups and Representations
- 10.1 Definition of a Group
- 10.2 Infinitesimal Ring or Lie Algebra
- 10.3 Lie Algebra of the Rotation Group
- 10.4 Group Invariants
- 10.5 Unitary Group in Two Dimensions
- 10.6 Lorentz Group and Its Lie Algebra
- 10.7 Group Representations
- 10.8 Representations of
- 10.9 Irreducible Representations of
- 10.10 Relation of and
- 10.11 Group Spaces
- 10.12 Hilbert Space and Quantum Mechanics
- 10.13 Continuous Groups and Symmetries
- Bibliography
- Problems
- Chapter 11: Complex Variables and Functions
-
Chapter 12: Complex Integrals and Series
- 12.1 Complex Integral Theorems
- 12.2 Taylor Series
- 12.3 Laurent Series
- 12.4 Classification of Singular Points
- 12.5 Residue Theorem
- 12.6 Analytic Continuation
- 12.7 Complex Techniques in Taking Some Definite Integrals
- 12.8 Gamma and Beta Functions
- 12.9 Cauchy Principal Value Integral
- 12.10 Integral Representations of Special Functions
- Bibliography
- Problems
-
Chapter 13: Fractional Calculus
- 13.1 Unified Expression of Derivatives and Integrals
- 13.2 Differintegrals
- 13.3 Other Definitions of Differintegrals
- 13.4 Properties of Differintegrals
- 13.5 Differintegrals of Some Functions
- 13.6 Mathematical Techniques with Differintegrals
- 13.7 Caputo Derivative
- 13.8 Riesz Fractional Integral and Derivative
- 13.9 Applications of Differintegrals in Science and Engineering
- Bibliography
- Problems
-
Chapter 14: Infinite Series
- 14.1 Convergence of Infinite Series
- 14.2 Absolute Convergence
- 14.3 Convergence Tests
- 14.4 Algebra of Series
- 14.5 Useful Inequalities About Series
- 14.6 Series of Functions
- 14.7 Taylor Series
- 14.8 Power Series
- 14.9 Summation of Infinite Series
- 14.10 Asymptotic Series
- 14.11 Method of Steepest Descent
- 14.12 Saddle-Point Integrals
- 14.13 Padé Approximants
- 14.14 Divergent Series in Physics
- 14.15 Infinite Products
- Bibliography
- Problems
-
Chapter 15: Integral Transforms
- 15.1 Some Commonly Encountered Integral Transforms
- 15.2 Derivation of the Fourier Integral
- 15.3 Fourier and Inverse Fourier Transforms
- 15.4 Conventions and Properties of the Fourier Transforms
- 15.5 Discrete Fourier Transform
- 15.6 Fast Fourier Transform
- 15.7 Radon Transform
- 15.8 Laplace Transforms
- 15.9 Inverse Laplace Transforms
- 15.10 Laplace Transform of a Derivative
- 15.11 Relation Between Laplace and Fourier Transforms
- 15.12 Mellin Transforms
- Bibliography
- Problems
-
Chapter 16: Variational Analysis
- 16.1 Presence of One Dependent and One Independent Variable
- 16.2 Presence of More than One Dependent Variable
- 16.3 Presence of More than One Independent Variable
- 16.4 Presence of Multiple Dependent and Independent Variables
- 16.5 Presence of Higher-Order Derivatives
- 16.6 Isoperimetric Problems and the Presence of Constraints
- 16.7 Applications to Classical Mechanics
- 16.8 Eigenvalue Problems and Variational Analysis
- 16.9 Rayleigh–Ritz Method
- 16.10 Optimum Control Theory
- 16.11 Basic Theory: Dynamics versus Controlled Dynamics
- Bibliography
- Problems
- Chapter 17: Integral Equations
- Chapter 18: Green's Functions
-
Chapter 19: Green's Functions and Path Integrals
- 19.1 Brownian Motion and the Diffusion Problem
- 19.2 Methods of Calculating Path Integrals
- 19.3 Path Integral Formulation of Quantum Mechanics
- 19.4 Path Integrals Over Lévy Paths and Anomalous Diffusion
- 19.5 Fox's -Functions
- 19.6 Applications of -Functions
- 19.7 Space Fractional Schrödinger Equation
- 19.8 Time Fractional Schrödinger Equation
- Bibliography
- Problems
- Further Reading
- Index
- End User License Agreement
Product information
- Title: Mathematical Methods in Science and Engineering, 2nd Edition
- Author(s):
- Release date: March 2018
- Publisher(s): Wiley
- ISBN: 9781119425397
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