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 “howto” 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 multidisciplinary 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 SturmLiouville theory
 Approaches secondorder 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 selfcontained threesemester course for curriculum, selfstudy, 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 SingleElectron 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 FourTensors
 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 SaddlePoint 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 HigherOrder 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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