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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Table of Contents
Cover
Title Page
Copyright
Preface
About the Book
About the Second Edition
Acknowledgments
Chapter 1: Legendre Equation and Polynomials
1.1 Second-Order Differential Equations of Physics
1.2 Legendre Equation
1.3 Legendre Polynomials
1.4 Associated Legendre Equation and Polynomials
1.5 Spherical Harmonics
Bibliography
Problems
Chapter 2: Laguerre Polynomials
2.1 Central Force Problems in Quantum Mechanics
2.2 Laguerre Equation and Polynomials
2.3 Associated Laguerre Equation and Polynomials
Bibliography
Problems
Chapter 3: Hermite Polynomials
3.1 Harmonic Oscillator in Quantum Mechanics
3.2 Hermite Equation and Polynomials
Bibliography
Problems
Chapter 4: Gegenbauer and Chebyshev Polynomials
4.1 Wave Equation on a Hypersphere
4.2 Gegenbauer Equation and Polynomials
4.3 Chebyshev Equation and Polynomials
Bibliography
Problems
Chapter 5: Bessel Functions
5.1 Bessel's Equation
5.2 Bessel Functions
5.3 Modified Bessel Functions
5.4 Spherical Bessel Functions
5.5 Properties of Bessel Functions
5.6 Transformations of Bessel Functions
Bibliography
Problems
Chapter 6: Hypergeometric Functions
6.1 Hypergeometric Series
6.2 Hypergeometric Representations of Special Functions
6.3 Confluent Hypergeometric Equation
6.4 Pochhammer Symbol and Hypergeometric Functions
6.5 Reduction of Parameters
Bibliography
Problems
Chapter 7: Sturm–Liouville Theory
7.1 Self-Adjoint Differential Operators
7.2 Sturm–Liouville Systems
7.3 Hermitian Operators
7.4 Properties of Hermitian Operators
7.5 Generalized Fourier Series
7.6 Trigonometric Fourier Series
7.7 Hermitian Operators in Quantum Mechanics
Bibliography
Problems
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
11.1 Complex Algebra
11.2 Complex Functions
11.3 Complex Derivatives and Cauchy–Riemann Conditions
11.4 Mappings
Bibliography
Problems
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
17.1 Classification of Integral Equations
17.2 Integral and Differential Equations
17.3 Solution of Integral Equations
17.4 Hilbert–Schmidt Theory
17.5 Neumann Series and the Sturm–Liouville Problem
17.6 Eigenvalue Problem for the Non-Hermitian Kernels
Bibliography
Problems
Chapter 18: Green's Functions
18.1 Time-Independent Green's Functions in One Dimension
18.2 Time-Independent Green's Functions in Three Dimensions
18.3 Time-Independent Perturbation Theory
18.4 First-Order Time-Dependent Green's Functions
18.5 Second-Order Time-Dependent Green's Functions
Bibliography
Problems
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
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Legendre Equation and Polynomials
Figure 1.1 Addition theorem.
Chapter 5: Bessel Functions
Figure 5.1 Flexible chain.
Figure 5.2 and functions.
Figure 5.3 Channel waves.
Figure 5.4 Bending of a rod.
Chapter 8: Factorization Method
Figure 8.1 Different cases for
Chapter 9: Coordinates and Tensors
Figure 9.1 Cartesian coordinate system.
Figure 9.2 Scalar and vector products.
Figure 9.3 Motion in Cartesian coordinates.
Figure 9.4 Unit basis vectors.
Figure 9.5 Angles for the direction cosines in .
Figure 9.6 Passive and active views of the rotation matrix.
Figure 9.7 In a general deformation, every point is displaced.
Figure 9.8 Hydrostatic compression.
Figure 9.9 Longitudinally stretched bar.
Figure 9.10 Pure shear.
Figure 9.11 Covariant and contravariant components.
Figure 9.12 Cartesian and plane polar coordinates.
Figure 9.13 Bugs living on a sphere.
Figure 9.14 Geometry is an experimental science.
Figure 9.15 Parallel transport.
Figure 9.16 Parallelogram.
Figure 9.17 A point in Minkowski spacetime.
Figure 9.18 Lorentz transformations.
Figure 9.19 Worldline and four-velocity .
Figure 9.20 Orientation of the axis with respect to the frame.
Figure 9.21 Orientation of the axis with respect to the frame.
Chapter 10: Continuous Groups and Representations
Figure 10.1 Rotation by about an axis along
Figure 10.2 Effect of on .
Figure 10.3 Boost and boost plus rotation.
Figure 10.4 Counterclockwise rotation of the physical system by about
Figure 10.5 Transformation to the -axis.
Figure 10.6 and the coordinates.
Figure 10.7 Definition of the angles in the addition theorem of the spherical harmonics.
Figure 10.8 Evaluation of
Figure 10.9 Multipole expansion.
Chapter 11: Complex Variables and Functions
Figure 11.1 A point in the complex -plane.
Figure 11.2 A point in the -plane.
Figure 11.3 It is not interesting to look at real functions as mappings.
Figure 11.4 Inversion maps circles to circles.
Figure 11.5 Inversion maps straight lines to circles.
Figure 11.6 Cut line ends at a branch point.
Figure 11.7 (a) For every point in every region that contains the origin has the full range , hence is multivalued. (b) For every region that does not include the origin, is single valued. (b)(f) For a single valued definition of the function , we extend the region in (b) without including the origin.
Figure 11.8 Each time we cross the cut line, changes from one branch value to another.
Figure 11.9 Riemann sheets for the function.
Figure 11.10 Cut lines for .
Figure 11.11 A different choice for the cut lines of
Figure 11.12 Branch cuts for Example 11.11.
Figure 11.13 Points below the real axis, which are symmetric to are respectively.
Figure 11.14 Angles in conformal mapping.
Figure 11.15 Two plates with hyperbolic cross sections.
Figure 11.16 Equipotentials and electric field lines in the -plane.
Figure 11.17 Two conductors with semicircular cross sections.
Figure 11.18 Two semicircular conductors in the
w
-plane.
Figure 11.19 Flow around a wall of height .
Figure 11.20 Transition from the -plane to the -plane.
Figure 11.21 Streamlines in the and -planes.
Figure 11.22 Mapping for Example 11.16.
Figure 11.23 Schwarz–Christoffel transformation maps the inside of a polygon to the upper half of the -plane.
Figure 11.24 Region we map in Example 11.17.
Figure 11.25 The polygon whose interior goes to the desired region in Example 11.17 in the limit
Figure 11.26 Semi-infinite parallel plate capacitor.
Figure 11.27 Limit of the point .
Figure 11.28 -Plane for the semi-infinite parallel plate capacitor.
Figure 11.29 -Plane for the semi-infinite parallel plate capacitor.
Figure 11.30 Equipotentials for the semi-infinite parallel plate capacitor.
Figure 11.31 Schwarz–Christoffel transformation for Example 11.19.
Figure 11.32 Schwarz–Christoffel transfomation for Example 11.20.
Figure 11.33 Two-dimensional equipotential problem.
Figure 11.34 Schwarz–Christoffel transformation.
Figure 11.35 Rectangular region surrounded by metallic plates.
Figure 11.36 Triangular region.
Figure 11.37 Conducting circular cylinder parallel to infinite metallic plate.
Chapter 12: Complex Integrals and Series
Figure 12.1 Contour for the Cauchy–Goursat theorem.
Figure 12.2 Contour for the Cauchy integral theorem.
Figure 12.3 The modified contour for the Cauchy integral theorem.
Figure 12.4 Contour for the Taylor series.
Figure 12.5 Laurent series are defined in an annular region.
Figure 12.6 Another contour for the Laurent series.
Figure 12.7 For the function, we write the Taylor series in the region .
Figure 12.8 For the function, we write the Laurent series in the region
Figure 12.9 In the residue theorem, a function has finite number of isolated singular points.
Figure 12.10 Contour for the residue theorem.
Figure 12.11 Contours for Example 12.6.
Figure 12.12 A function with isolated singular points at , , and
Figure 12.13 Analytic continuation by successive Taylor series expansions.
Figure 12.14 Contour for the type II integrals.
Figure 12.15 Contour for Example 12.9.
Figure 12.16 Contour in the limit for type III integrals.
Figure 12.17 Upper bound calculation for type III integrals.
Figure 12.18 Contour for the type IV integrals.
Figure 12.19 Contour for the Hankel definition of
Figure 12.20 Contour for the Cauchy principal value integral.
Figure 12.21 Another contour for the Cauchy principal value calculation.
Figure 12.22 Contour for
Figure 12.23 Contour for
Figure 12.24 Contour for the Schläfli formula of Legendre polynomials.
Figure 12.25 Contour for
Figure 12.26 Contour for
Figure 12.27
Figure 12.28 For the integer values , there is no need for the branch cut; hence, the contour for the integral definition of can be deformed into
Figure 12.29 The contour for , where takes integer values, can be taken as any closed path enclosing the origin.
Figure 12.30 Contour for problem 5.
Chapter 13: Fractional Calculus
Figure 13.1 Contour for the Cauchy integral formula.
Figure 13.2 Contour in the differintegral formula.
Figure 13.3 Contour in the differintegral formula.
Figure 13.4 Contours for the and integrals.
Figure 13.5 Mittag–Leffler functions.
Figure 13.6 Random walk and CTRW.
Figure 13.7 Probability distribution in random walk and CTRW.
Figure 13.8 Evolution of the probability distribution with position and time in arbitrary units for the harmonic oscillator potential.
Chapter 14: Infinite Series
Figure 14.1 Integral test.
Figure 14.2 Uniform convergence is very important.
Figure 14.3 Contour for finding series sums using the residue theorem.
Figure 14.4 In one dimension, the method of steepest descent allows us to approximate the integrand in Eq. (14.190), that has a high maximum at with a Gaussian, where the width, is and
Figure 14.5 The path is the path that follows the steepest descent. The path is perpendicular to , hence it follows the ridges.
Figure 14.6 Two possible
mountain ranges
: For the one on the left, we use and for the one on the right, we use in Eq. (14.235).
Chapter 15: Integral Transforms
Figure 15.1 Wave train with
Figure 15.2 function.
Figure 15.3 A narrow beam going through a homogeneous material of thickness .
Figure 15.4 Reference axes and , projection angle and the detector plane .
Figure 15.5 Projection of onto the detector surface .
Figure 15.6 Null function.
Figure 15.7 Heaviside step function.
Figure 15.8 Nutation of Earth.
Figure 15.9 Pendulums connected by a spring.
Chapter 16: Variational Analysis
Figure 16.1 Variation of paths.
Figure 16.2 Soap film between two rings.
Figure 16.3 Newton's bucket experiment with a cylindrical container.
Figure 16.4 Drag force on a surface of revolution.
Figure 16.5 Deformation of an elastic beam.
Figure 16.6 could be approximated by
Chapter 18: Green's Functions
Figure 18.1 Contour for Case I:
Figure 18.2 Contour for Case I:
Figure 18.3 Contours for Case II.
Figure 18.4 Contours for Case III.
Figure 18.5 Contours for Case IV.
Figure 18.6 Contours for Case V.
Figure 18.7 Contours for the harmonic oscillator.
Figure 18.8 Diffraction from a circular aperture.
Figure 18.9 Gaussian.
Chapter 19: Green's Functions and Path Integrals
Figure 19.1 Paths for the pinned Wiener measure.
Figure 19.2 Paths for the unpinned Wiener measure.
Figure 19.3 Paths for the time slice method.
Figure 19.4 Path and deviation in the “semiclassical” method.
Figure 19.5 Rotation by in the complex- plane.
Figure 19.6 Modified Bromwich contour.
Mathematical Methods in Science and Engineering
Selçuk Ş. Bayın
Institute of Applied Mathematics Middle East Technical University Ankara Turkey
Second Edition
This edition first published 2018
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Library of Congress Cataloguing-in-Publication Data:
Names: Bayin, Ş. Selçuk, 1951- author.
Title: Mathematical methods in science and engineering / by Selçuk Ş. Bayin.
Description: Second edition. | Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |
Identifiers: LCCN 2017042888 (print) | LCCN 2017048224 (ebook) | ISBN 9781119425410 (pdf) | ISBN 9781119425458 (epub) | ISBN 9781119425397 (cloth)
Subjects: LCSH: Mathematical physics-Textbooks. | Engineering mathematics-Textbooks.
Classification: LCC QC20 (ebook) | LCC QC20 .B35 2018 (print) | DDC 530.15-dc23
LC record available at https://lccn.loc.gov/2017042888
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Preface
Courses on mathematical methods of physics are among the essential courses for graduate programs in physics, which are also offered by most engineering departments. Considering that the audience in these courses comes from all subdisciplines of physics and engineering, the content and the level of mathematical formalism has to be chosen very carefully. Recently, the growing interest in interdisciplinary studies has brought scientists together from physics, chemistry, biology, economy, and finance and has increased the demand for these courses in which upper-level mathematical techniques are taught. It is for this reason that the mathematics departments, who once overlooked these courses, are now themselves designing and offering them.
Most of the available books for these courses are written with theoretical physicists in mind and thus are somewhat insensitive to the needs of this new multidisciplinary audience. Besides, these books should not only be tuned to the existing practical needs of this multidisciplinary audience but should also play a lead role in the development of new interdisciplinary science by introducing new techniques to students and researchers.
About the Book
We give a coherent treatment of the selected topics with a style that makes advanced mathematical tools accessible to a multidisciplinary audience. The book is written in a modular way so that each chapter is actually a review of its subject and can be read independently. This makes the book very useful not only as a self-study book for students and beginning researchers but also as a reference for scientists. We emphasize physical motivation and the multidisciplinary nature of the methods discussed. Whenever possible, we prefer to introduce mathematical techniques through physical applications. Examples are often used to extend discussions of specific techniques rather than as mere exercises.
Topics are introduced in a logical sequence and discussed thoroughly. Each sequence climaxes with a part where the material of the previous chapters is unified in terms of a general theory, as in Chapter 7 on the Sturm–Liouville theory, or as in Chapter 18 on Green's functions, where the gains of the previous chapters are utilized. Chapter 8 is on factorization method. It is a natural extension of our discussion on the Sturm–Liouville theory. It also presents a different and an advanced treatment of special functions. Similarly, Chapter 19 on path integrals is a natural extension of our chapter on Green's functions. Chapters 9 and 10 on coordinates, tensors, and continuous groups have been located after Chapter 8 on the Sturm–Liouville theory and the factorization method. Chapters 11 and 12 are on complex techniques, and they are self-contained. Chapter 13 on fractional calculus can either be integrated into the curriculum of the mathematical methods of physics courses or used independently to design a one-semester course.
Since our readers are expected to be at least at the graduate or the advanced undergraduate level, a background equivalent to the contents of our undergraduate text book Essentials of Mathematical Methods in Science and Engineering (Bayin, 2008) is assumed. In this regard, the basics of some of the methods discussed here can be found there. For communications about the book, we will use the website http://www.users.metu.edu.tr/bayin/
The entire book contains enough material for a three-semester course meeting three hours a week. The modular structure of the book gives enough flexibility to adopt the book for two- or even a one-semester course. Chapters 1–7, 11, 12, and 14–18 have been used for a two-semester compulsory graduate course meeting three hours a week at METU, where students from all subdisciplines of physics meet. In other universities, colleagues have used the book for their two or one semester courses.
During my lectures and first reading of the book, I recommend that readers view equations as statements and concentrate on the logical structure of the arguments. Later, when they go through the derivations, technical details will be understood, alternate approaches will appear, and some of the questions will be answered. Sufficient numbers of problems are given at the back of each chapter. They are carefully selected and should be considered an integral part of the learning process. Since some of the problems may require a good deal of time, we recommend the reader to skim through the entire problem section before attempting them. Depending on the level and the purpose of the reader, certain parts of the book can be skipped in first reading. Since the modular structure of the book makes it relatively easy for the readers to decide on which chapters or sections to skip, we will not impose a particular selection.
In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics. In fact, the number of books, old and new, written on this subject shows how dynamic this field is. Naturally, this book carries an imprint of my style and lectures. Because the main aim of this book is pedagogy, occasionally I have followed other books when their approaches made perfect sense to me. Main references are given at the back of each chapter. Additional references can be found at the back. Readers of this book will hopefully be well prepared for advanced graduate studies and research in many areas of physics. In particular, as we use the same terminology and style, they should be ready for full-term graduate courses based on the books: The Fractional Calculus by Oldham and Spanier and Path Integrals in Physics, Volumes I and II by Chaichian and Demichev, or they could jump to the advanced sections of these books, which have become standard references in their fields. Our list of references, by all means, is not meant to be complete or up to date. There are many other excellent sources that nowadays the reader can locate by a simple internet search. Their exclusion here is simply ignorance on my part and not a reflection on their quality or importance.
About the Second Edition
The challenge in writing a mathematical methods text book is that for almost every chapter an entire book can be devoted. Sometimes, even sections could be expanded into another book. In this regard, it is natural that books with such broad scope need at least another edition to settle down. The second edition of Mathematical Methods in Science and Engineering corresponds to a major overhaul of the entire book. In addition to 34 new examples, 34 new figures, and 48 new problems, over 60 new sections/subsections have been included on carefully selected topics that make the book more appealing and useful to its multidisciplinary audience.
Among the new topics introduced, we have the discrete and fast Fourier transforms; Cartesian tensors and the theory of elasticity; curvature; Caputo and Riesz fractional derivatives; method of steepest descent and saddle-point integrals; Padé approximants; Radon transforms; optimum control theory and controlled dynamics; diffraction; time independent perturbation theory; the anharmonic oscillator problem; anomalous diffusion; Fox's H-functions and many others. As Socrates has once said education is the kindling of a flame, not the filling of a Vessel, all topics are selected and written, not to fill a vessel but to inform, provoke further thought, and interest among the multidisciplinary audience we address.
Besides these, throughout the book, countless changes have been made to assure easy reading and smooth flow of the complex mathematical arguments. Derivations are given in sufficient detail so that the reader will not be distracted by searching for results in other parts of the book or by needing to write down equations. We have shown carefully selected keywords in boldface and framed key results so that information can be located easily as the reader scans through the pages. Also, using the new Wiley style and a more efficient way of displaying equations, we were able to keep the book at an optimum size.
Acknowledgments
I would again like to start by paying tribute to all the scientists and mathematicians whose works contributed to the subjects discussed in this book. I would also like to compliment the authors of the existing books on mathematical methods of physics. I appreciate the time and dedication that went into writing them. Most of them existed even before I was a graduate student and I have benefitted from them greatly. As in the first edition, I am indebted to Prof. K. T. Hecht of the University of Michigan, whose excellent lectures and clear style had a great influence on me. I am grateful to Prof. P. G. L. Leach for sharing his wisdom with me and for meticulously reading Chapters 8, 13, and 19. I also thank Prof. N. K. Pak for many interesting and stimulating discussions, encouragement, and critical reading of the chapter on path integrals. Their comments kept illuminating my way during the preparation of this edition as well. I thank Prof. E. Akyıldız and Prof. B. Karasözen for encouragement and support at the Institute of Applied Mathematics at METU, which became home to me. I also thank my editors Jon Gurstelle and Kathleen Pagliaro, and the publication team at Wiley for sharing my excitement and their utmost care in bringing this book into existence. Finally, I thank my beloved wife Adalet and darling daughter Sumru. Without their endless love and support, this project, which spanned over a decade, would not have been possible.
Selçuk Ş. Bayın
METU/IAM Ankara/TURKEY July 2017
Chapter 1Legendre Equation and Polynomials
Legendre polynomials, are the solutions of the Legendre equation:
They are named after the French mathematician Adrien-Marie Legendre (1752–1833). They are frequently encountered in physics and engineering applications. In particular, they appear in the solutions of the Laplace equation in spherical polar coordinates.
1.1 Second-Order Differential Equations of Physics
Many of the second-order partial differential equations of physics and engineering can be written as
where some of the frequently encountered cases are:
1.
When
and
are zero, we have the
Laplace equation
:
which is encountered in many different areas of science like electrostatics, magnetostatics, laminar (irrotational) flow, surface waves, heat transfer and gravitation.
2.
When the right-hand side of the Laplace equation is different from zero, we have the
Poisson equation
:
where
represents sources or sinks in the system.
3.
The
Helmholtz wave equation
is written as
where
is a constant.
4.
Another important example is the time-independent
Schrödinger equation
:
where
in Eq. (
1.2
) is zero and
is given as
A common property of all these equations is that they are linear and second-order partial differential equations. Separation of variables, Green's functions and integral transforms are among the frequently used analytic techniques for obtaining solutions. When analytic methods fail, one can resort to numerical techniques like Runge–Kutta. Appearance of similar differential equations in different areas of science allows one to adopt techniques developed in one area into another. Of course, the variables and interpretation of the solutions will be very different. Also, one has to be aware of the fact that boundary conditions used in one area may not be appropriate for another. For example, in electrostatics, charged particles can only move perpendicular to the conducting surfaces, whereas in laminar (irrotational) flow, fluid elements follow the contours of the surfaces; thus even though the Laplace equation is to be solved in both cases, solutions obtained in electrostatics may not always have meaningful counterparts in laminar flow.
1.2 Legendre Equation
We now solve Eq. (1.2) in spherical polar coordinates using the method of separation of variables. We consider cases where is only a function of the radial coordinate and also set to zero. The time-independent Schrödinger equation (1.6) for the central force problems, is an important example for such cases. We first separate the radial, and the angular variables and write the solution as . This basically assumes that the radial dependence of the solution is independent of the angular coordinates and vice versa. Substituting this in Eq. (1.2), we get
After multiplying by and collecting the dependence on the right-hand side, we obtain
Since and are independent variables, this equation can be satisfied for all and only when both sides of the equation are equal to the same constant. We show this constant with , which is also called the separation constant. Now Eq. (1.9) reduces to the following two equations:
where Eq. (1.10) for is an ordinary differential equation. We also separate the and the variables in as and call the new separation constant , and write
The differential equations to be solved for and are now found, respectively, as
In summary, using the method of separation of variables, we have reduced the partial differential equation [Eq. (1.8)] to three ordinary differential equations [Eqs. (1.10), (1.13), and (1.14)]. During this process, two constant parameters, and called the separation constants have entered into our equations, which so far have no restrictions on them.
1.2.1 Method of Separation of Variables
In the above discussion, the fact that we are able to separate the solution is closely related to the use of the spherical polar coordinates, which reflect the symmetry of the central force problem, where the potential, depends only on the radial coordinate. In Cartesian coordinates, the potential would be written as and the solution would not be separable as Whether a given partial differential equation is separable or not is closely linked to the symmetries of the physical system. Even though a proper discussion of this point is beyond the scope of this book, we refer the reader to [9] and suffice by saying that if a partial differential equation is not separable in a given coordinate system, it is possible to check the existence of a coordinate system in which it would be separable. If such a coordinate system exists, then it is possible to construct it from the generators of the symmetries.
Among the three ordinary differential equations [Eqs. (1.10), (1.13), and (1.14)], Eq. (1.14) can be solved immediately with the general solution
where the separation constant, is still unrestricted. Imposing the periodic boundary condition we restrict to integer values: Note that in anticipation of applications to quantum mechanics, we have taken the two linearly independent solutions as For the other problems, and could be used.
For the differential equation to be solved for [Eq. (1.13)], we define a new independent variable, and write
For , this equation is called the Legendre equation. For , it is known as the associated Legendre equation.
1.2.2 Series Solution of the Legendre Equation
Starting with the case, we write the Legendre equation as
This has two regular singular points at and 1. Since these points are at the end points of our interval, we use the Frobenius method [8] and try a series solution about the regular point as , where is a constant. Substituting this into Eq. (1.17), we get
We now write the first two terms of the first series explicitly:
and make the variable change to write Eq. (1.18) as
From the uniqueness of power series, this equation cannot be satisfied for all unless the coefficients of all the powers of vanish simultaneously. This gives the following relations among the coefficients:
Equation (1.21), which is obtained by setting the coefficient of the lowest power of to zero, is called the indicial equation. Assuming , the two roots of the indicial equation give the values and while the remaining Eqs. (1.22) and (1.23) give the recursion relation among the coefficients.
Starting with the root we write
and obtain the remaining coefficients as
Since Eq. (1.22) with implies all the odd coefficients vanish, thus yielding the following series solution for :
For the other root, Eqs. (1.21) and (1.22) imply and thus the recursion relation:
determines the nonzero coefficients as
Now the series solution for is obtained as
The Legendre equation is a second-order linear ordinary differential equation, which in general has two linearly independent solutions. Since and are arbitrary, we note that the solution for also contains the solution for ; hence the general solution can be written as
where and are two integration constants to be determined from the boundary conditions. These series are called the Legendre series.
1.2.3 Frobenius Method – Review
A second-order linear homogeneous ordinary differential equation with two linearly independent solutions may be put in the form
If is no worse than a regular singular point, that is, when
and
we can seek a series solution of the form
Substituting this series into the above differential equation and setting the coefficient of the lowest power of with gives us a quadratic equation for called the indicial equation. For almost all the physically interesting cases, the indicial equation has two real roots. This gives us the following possibilities for the two linearly independent solutions of the differential equation [8]:
1.
If the two roots
differ by a noninteger, then the two linearly independent solutions,
and
are given as
2.
If
where
and
is a positive integer, then the two linearly independent solutions,
and
are given as
The second solution contains a logarithmic singularity, where
is a constant that may or may not be zero. Sometimes,
will contain both solutions; hence it is advisable to start with the smaller root with the hopes that it might provide the general solution.
3.
If the indicial equation has a double root,
then the Frobenius method yields only one series solution. In this case, the two linearly independent solutions can be taken as
where the second solution diverges logarithmically as
In the presence of a double root, the Frobenius method is usually modified by taking the two linearly independent solutions,
and
as
In all these cases, the general solution is written as
1.3 Legendre Polynomials
Legendre series are convergent in the interval . This can be checked easily by the ratio test. To see how they behave at the end points, we take the limit of the recursion relation in Eq. (1.30) to obtain For sufficiently large values, this means that both series behave as
The series inside the parentheses is nothing but the geometric series:
Hence both of the Legendre series diverge at the end points as . However, the end points correspond to the north and the south poles of a sphere. Because the problem is spherically symmetric, there is nothing special about these points. Any two diametrically opposite points can be chosen to serve as the end points. Hence we conclude that the physical solution should be finite everywhere on a sphere. To avoid the divergence at the end points we terminate the Legendre series after a finite number of terms. This is accomplished by restricting the separation constant to integer values:
With this restriction on , one of the Legendre series in Eq. (1.33) terminates after a finite number of terms while the other one still diverges at the end points. Choosing the coefficient of the divergent series in the general solution as zero, we obtain the polynomial solutions of the Legendre equation as
These polynomials are called the Legendre polynomials, which are finite everywhere on a sphere. They are defined so that their value at is one. In general, they can be expressed as
where means the greatest integer in the interval . Restriction of to certain integer values for finite solutions everywhere is a physical (boundary) condition and has very significant physical consequences. For example, in quantum mechanics, it means that magnitude of the angular momentum is quantized. In wave mechanics, like the standing waves on a string fixed at both ends, it means that waves on a sphere can only have certain wavelengths.
1.3.1 Rodriguez Formula
Another definition of the Legendre polynomials is given by the Rodriguez formula:
To show that this is equivalent to the previous definition in Eq. (1.49), we use the binomial formula [4]:
to write Eq. (1.51) as
We now use the formula
to obtain
thus proving the equivalence of Eqs. (1.51) and (1.49).
1.3.2 Generating Function
Another way to define the Legendre polynomials is using a generating function, , which is given as
To show that generates the Legendre polynomials, we write as
and use the binomial expansion
Deriving the useful relation:
we write Eq. (1.58) as
which after substituting in Eq. (1.57) gives
Employing the binomial formula once again to expand the factor
