Undergraduate Lecture Notes in Physics

Home , Giacinto Morera, Josip Plemelj, Mechanician

More information about this series at http://www.springer.com/series/8917 Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing prac- tice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: An exceptionally clear and concise treatment of a standard undergraduate subject. A solid undergraduate-level introduction to a graduate, advanced, or nonstandard subject. A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career.

Series editors

Neil Ashby Professor Emeritus, University of Colorado Boulder, CO, USA

William Brantley Professor, Furman University, Greenville, SC, USA

Matthew Deady Professor, Bard College, Annandale, NY, USA

Michael Fowler Professor, University of Virginia, Charlottesville, VA, USA

Morten Hjorth-Jensen Professor, University of Oslo, Norway

Michael Inglis Professor, SUNY Suffolk County Community College, Selden, NY, USA

Heinz Klose Professor Emeritus, Humboldt University Berlin, Germany

Helmy Sherif Professor, University of Alberta, Edmonton, AB, Canada Lev Kantorovich

Mathematics for Natural Scientists II Advanced Methods

123 Lev Kantorovich Physics Department School of Natural and Mathematical Sciences King’s College London, The Strand London, UK

ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notes in Physics ISBN 978-3-319-27859-9 ISBN 978-3-319-27861-2 (eBook) DOI 10.1007/978-3-319-27861-2

Library of Congress Control Number: 2015943266

© Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Sprinter imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface

This is the second volume of the course of mathematics for natural scientists. It is loosely based on the mathematics course for second year physics students at King’s College London that I have been reading for more than 10 years. It follows the spirit of the first volume [1] by continuing a gradual build-up of the mathematical knowledge necessary for, but not exclusively, physics students. This volume covers more advanced material, beginning with two essential components: linear algebra (Chap. 1) and theory of functions of complex variables (Chap. 2). These techniques are heavily used in the chapters that follow. Fourier series are considered in Chap. 3, special functions of mathematical physics (Dirac delta function, gamma and beta functions, detailed treatment of orthogonal polynomials, the hypergeometric differential equation, spherical and Bessel functions) in Chap. 4 and then Fourier (Chap. 5) and Laplace (Chap. 6) transforms. In Chap. 7, a detailed treatment of curvilinear coordinates is given, including the corresponding differential calculus. This is essential, as many physical problems possess symmetry and using appropriate curvilinear coordinates may significantly simplify the solution of the corresponding partial differential equations (studied in Chap. 8) if the symmetry of the problem at hand is taken into account. The book is concluded with variational calculus in Chap. 9. As in the first volume, I have tried to introduce new concepts gradually and as clearly as possible, giving examples and problems to illustrate the material. Across the text, all the proofs necessary to understand and appreciate the mathematics involved are also given. In most cases, the proofs would satisfy the most demanding physicist or even a mathematician; only in a few cases have I had to sacrifice the “strict mathematical rigour” by presenting somewhat simplified derivations and/or proofs. As in the first volume, many problems are given throughout the text. These are designed mainly to illustrate the theoretical material and require the reader to complete them in order to be in a position to move forward. In addition, other problems are offered for practise, although I have to accept, their number could have been larger. For more problems, the reader is advised to consult other texts, e.g. the books [2–6].

v vi Preface

When working on this volume, I have mostly consulted a number of excellent classic Russian textbooks [7–12]. As far as I am aware, some of them are available in English, and I would advise a diligent student to continue his/her education by reading these. Concerning the others, there is of course an obvious language barrier. Unfortunately, as I cannot ask the reader to learn Russian purely for that purpose, these texts remain inaccessible for most readers. I hope that the reader would be able to find more specialised texts in English, which go beyond the scope of this (and the previous) book to further the development of their studies, e.g. books [13– 26] represent a rather good selection which cover the topics of this volume, but this list of course by no means is complete. The mathematics throughout the book is heavily illustrated by examples from condensed matter physics. In fact, probably over a quarter of the text of the whole volume is occupied by these. Every chapter, save Chap. 8, contains a large concluding section exploring physics topics that necessitate the mathematics presented in that chapter. Chapter 8, on partial differential equations, is somewhat special in this respect, as it is entirely devoted to solving equations of mathematical physics (wave, Laplace and heat transport equations). Consequently, it does not have a special section on applications. When selecting the examples from physics, I was mostly governed by my own experience and research interests as well as several texts, such as the books [27, 28]. In fact, examples from [27] have been used in both of these volumes. As in the first volume, this book begins with a list of the names of all the scientists across the world, mathematicians, physicists and engineers, whose invaluable contribution formed the foundation of the beautiful sciences of mathematics and physics that have been enjoying a special bond throughout the centuries. Should you find any errors or unnoticed misprints, please send your corrections either directly to myself ([email protected]) or to the publisher. Your general comments, suggestions and any criticism related to these two volumes would be greatly appreciated. This book concludes the project I started about four years ago, mainly working in the evenings, at the weekends, as well as on the train going to and from work. Not everything I initially planned to include has appeared in the books, although the two volumes contain most of the essential ideas young physicists, engineers and computational chemists should become familiar with. Theory of operators, group theory, tensor calculus, stochastic theory (or theory of probabilities) and some other more specialised topics have not been included, but can be found in multiple other texts. Still, I hope these volumes will serve as an enjoyable introduction to the beautiful world of mathematics for students, encouraging them to think more and ask for more. I am also confident that the books will serve as a rich source for lecturers. I wish you happy reading!

London, UK Lev Kantorovich Preface vii

References

1. L. Kantorovich, “Mathematics for Natural Scientists: Fundamentals and Basics”, Undergradu- ate Lecture Notes in Physics, Springer, 2015 (ASIN: B016CZWP36). 2. D. McQuarrie, “Mathematical methods for scientists and engineers”, Univ. Sci. Books, 2003 (ISBN 1-891389-29-7). 3. K. F. Riley, M. P. Hobson, and S. J. Bence, “Mathematical Methods for Physics and Engineering”, Cambridge Univ. Press, 2006 (ISBN 0521679710). 4. M. Boas, “Mathematical methods in the physical sciences”, Wiley, 2nd Edition, 1983 (ISBN 0-471-04409-1). 5. K. Stroud, “Engineering mathematics”, Palgrave, 5th Edition, 2001 (ISBN 0-333-919394). 6. G. B. Arfken, H. J. Weber and F. E. Harris, “Mathematical Methods for Physicists. A Comprehensive Guide”, Academic Press, 7th Edition, 2013 (ISBN: 978-0-12-384654-9). 7. В. И. Смирнов,“Курс высшей математики”, т.1-5,Москва, Наука, 1974 (in Russian). Apparently, there is a rather old English translation: V. I. Smirnov, “A Course of Higher Mathematics: Adiwes International Series in Mathematics”, Pergamon, 1964; Vols. 1 (ISBN 1483123944), 2 (ISBN 1483120171), 3-Part-1 (ISBN B0007ILX1K) and 3-Part-2 (ISBN B00GWQPPMO). 8. Г. М. Фихтенголц,“Курс дифференциального и интегрального исчисления”, Москва, Физматлит, 2001 (in Russian). [G. M. Fihtengolc, “A Course of Differentiation and Integration”, Vols. 1–3., Fizmatgiz, 2001] I was not able to find an English translation of this book. 9. В. С. Шипачев,“Высшая математика”, Москва, Высшая Школа, 1998 (in Russian). [V. S. Shipachev, “Higher Mathematics”, Vishaja Shkola, 1998.] I was not able to find an English translation of this book. 10. А. Н. Тихонов и А. А. Самарский,“Уравнения математической физики”, Москва, Наука, 1977 (in Russian). There is a resent reprint of a rather old translation: A. N. Tikhonov and A. A. Samarskii, “Equations of Mathematical Physics”, Dover Books on Physics, 2011 (ISBN-10: 0486664228, ISBN-13: 978-0486664224). 11. М. А. Лаврентьев и Б. В. Шабат,“Методы теории функций комплексного переменного”,УМН, 15:5(95), 1960 (in Russian). [M. A. Lavrent’ev and B. V. Shabat, Methods of theory of functions of complex variables”, Uspekhi Mat. Nauk, 15:5(95) (1960).] I was not able to find an English translation of this book. 12. А. Ф. Никофоров и В. Б. Уваров,“Специальные функции математической физики”, Москва, Наука, 1984 (in Russian). This excellent book is available in English: A. F. Nikiforov and V. B. Uvarov, “Special Functions of Mathematical Physics: A Unified Introduction With Applications”, Springer, 2013. (ISBN-10: 1475715978, ISBN-13: 978-1475715972). 13. J. M. Howie, “Complex analysis”, Springer, 2004 (ISBN-10: 1852337338, ISBN-13: 978- 1852337339). 14. H. A. Priestley, “Introduction to Complex Analysis Paperback”, OUP Oxford; 2 edition, 2003 (ISBN-10: 0198525621, ISBN-13: 978-0198525622) 15. Kevin W. Cassel, “Variational Methods with Applications in Science and Engineering Hardcover”, Cambridge University Press, 2013 (ISBN-10: 1107022584, ISBN-13: 978- 1107022584). 16. Stanley J. Farlow , “Partial Differential Equations for Scientists and Engineers”, Dover Books on Mathematics, Dover, 2003 (ISBN-10: 048667620X, ISBN-13: 978-0486676203). 17. Geoffrey Stephenson, “Partial Differential Equations for Scientists and Engineers”, Imperial College Press, 1996 (ISBN-10: 1860940242, ISBN-13: 978-1860940248). 18. Phil Dyke, “An Introduction to Laplace Transforms and Fourier Series”, Springer Undergradu- ate Mathematics Series, Springer, 2014 (ISBN-10: 144716394X, ISBN-13: 978-1447163947). 19. Brian Davies, “Integral Transforms and Their Applications”, Texts in Applied Mathematics, Springer, 2002 (ISBN-10: 0387953140, ISBN-13: 978-0387953144). viii Preface

20. Allan Pinkus, “Fourier Series and Integral Transforms”, Cambridge University Press, 1997 (ISBN-10: 0521597714, ISBN-13: 978-0521597715). 21. Serge Lang, “Linear Algebra”, Undergraduate Texts in Mathematics, Springer, 2013 (ISBN-10: 1441930817, ISBN-13: 978-1441930811). 22. Richard Bellman, “Introduction to Matrix Analysis”, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, 2 edition, 1987 ( ISBN-10: 0898713994, ISBN-13: 978-0898713992). 23. George Andrews, Richard Askey and Ranjan Roy, “Special Functions”, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2010 (ISBN-10: 0521789885, ISBN-13: 978-0521789882). 24. N. N. Lebedev, “Special Functions & Their Applications”, Dover Books on Mathematics, Dover, 2003 (ISBN-10: 0486606244, ISBN-13: 978-0486606248). 25. Naismith Sneddon, “Special Functions of Mathematical Physics and Chemistry”, Univer- sity Mathematics Texts, Oliver & Boyd, 1966 (ISBN-10: 0050013343, ISBN-13: 978- 0050013342). 26. W. W. Bell, “Special Functions for Scientists and Engineers”, Dover Books on Mathematics, Dover, 2004 (ISBN-10: 0486435210, ISBN-13: 978-0486435213). 27. В. Г. Левич,“Курс теоретической физики”, т.1и 2, Наука, 1969 (in Russian). There is a translation: Benjamin G. Levich, “Theoretical physics: an advanced text”, North-Holland Publishing in Amsterdam, London, 1970. 28. L. Kantorovich, “Quantum Theory of the Solid State: an Introduction”, Springer, 2004 (ISBN 978-1-4020-1821-3 and 978-1-4020-2153-4). Famous Scientists Mentioned in the Book

Throughout the book various people, both mathematicians and physicists, who are remembered for their outstanding contribution in developing science, will be mentioned. For reader’s convenience, their names (together with some information borrowed from their Wikipedia pages) are listed here in the order they ﬁrst appear in the text: Leopold Kronecker (1823–1891) was a German mathematician. Georg Friedrich Bernhard Riemann (1826–1866) was an inﬂuential German mathematician who made lasting and revolutionary contributions to analysis, number theory and differential geometry. Jørgen Pedersen Gram (1850–1916) was a Danish actuary and mathematician. Erhard Schmidt (1876–1959) was an Estonian-German mathematician. Adrien-Marie Legendre (1752–1833) was a French mathematician. Edmond Nicolas Laguerre (1834–1886) was a French mathematician. Charles Hermite (1822–1901) was a French mathematician. Pafnuty Lvovich Chebyshev (1821–1894) was a Russian mathematician. Albert Einstein (1879–1955) was a German-born theoretical physicist who developed the general theory of relativity and also contributed in many other areas of physics. He received the 1921 Nobel Prize in physics for his “services to theoretical physics”. Wolfgang Ernst Pauli (1900–1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. Jean-Baptiste Joseph Fourier (1768–1830) was a French mathematician and physicist. Gabriel Cramer (1704–1752) was a Swiss mathematician. Hendrik Antoon Lorentz (1853–1928) was a Dutch physicist. Józef Maria Hoene-Wronski´ (1776–1853) was a Polish Messianist philosopher, mathematician, physicist, inventor, lawyer and economist. Ludwig Otto Hesse (1811–1874) was a German mathematician. Cornelius (Cornel) Lanczos (1893–1974) was a Hungarian mathematician and physicist. Léon Nicolas Brillouin (1889–1969) was a French physicist.

ix x Famous Scientists Mentioned in the Book

George Green (1793–1841) was a British mathematical physicist. Erwin Rudolf Josef Alexander Schrödinger (1887–1961) was a Nobel Prize- winning (1933) Austrian physicist who developed a number of fundamental results, which formed the basis of wave mechanics. Sir William Rowan Hamilton (1805–1865) was an Irish physicist, astronomer and mathematician. Paul Adrien Maurice Dirac (1902–1984) was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. He shared the Nobel Prize in physics for 1933 with Erwin Schrödinger “for the discovery of new productive forms of atomic theory”. Abraham de Moivre (1667–1754) was a French mathematician. Giacinto Morera (1856–1909) was an Italian engineer and mathematician. Baron Augustin-Louis Cauchy (1789–1857) was a French mathematician widely reputed as a pioneer of analysis. Pierre-Simon, marquis de Laplace (1749–1827) was an inﬂuential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. Leonhard Euler (1707–1783) was a pioneering Swiss mathematician and physicist. Sir Isaac Newton (1642–1726/1727) was a famous English physicist and mathematician who laid the foundations for classical mechanics and made seminal contributions to optics and (together with Gottfried Leibniz) the development of calculus. Gottfried Wilhelm von Leibniz (1646–1716) was a German polymath and philosopher, who to this day occupies a prominent place in the history of mathematics and the history of philosophy. Most scholars believe Leibniz developed calculus independently of Isaac Newton, and Leibniz’s notation has been widely used ever since it was published. Karl Theodor Wilhelm Weierstrass (1815–1897) was a German mathematician often cited as the “father of modern analysis”. Niels Henrik Abel (1802–1829) was a Norwegian mathematician. Brook Taylor (1685–1731) was an English mathematician. Pierre Alphonse Laurent (1813–1854) was a French mathematician. Julian Karol Sochocki (1842–1927) was a Polish mathematician. Felice Casorati (1835–1890) was an Italian mathematician. Marie Ennemond Camille Jordan (1838–1922) was a French mathematician. Hendrik Anthony Kramers (1894–1952) was a Dutch physicist. Ralph Kronig (1904–1995) was a German-American physicist. Max Karl Ernst Ludwig Planck (1858–1947) was a German theoretical physicist and one of the founders of the quantum theory. Gerd Binnig (born 1947) is a German physicist. Heinrich Rohrer (born 1933) is a Swiss physicist. Julian Karol Sokhotski (1842–1927) was a Russian-Polish mathematician. Famous Scientists Mentioned in the Book xi

Josip Plemelj (1873–1967) was a Slovene mathematician. Oliver Heaviside (1850–1925) was a self-taught English electrical engineer, mathematician and physicist. Johann Carl Friedrich Gauss (1777–1855) was a German mathematician and physicist. Lorenzo Mascheroni (1750–1800) was an Italian mathematician. Benjamin Olinde Rodrigues (1795–1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician and social reformer. Carl Gustav Jacob Jacobi (1804–1851) was a German mathematician. Friedrich Wilhelm Bessel (1784–1846) was a German astronomer, mathematician, physicist and geodesist. James Stirling (1692–1770) was a Scottish mathematician. William Lawrence Bragg (1890–1971) was an Australian-born British physicist and X-ray crystallographer, discoverer (1912) of the Bragg law of X-ray diffraction and a joint winner (with his father, Sir William Bragg) of the Nobel Prize for physics in 1915. Niels Henrik David Bohr (1885–1962) was a Danish physicist who made fundamental contributions to quantum theory. He received the Nobel Prize in physics in 1922. Ludwig Eduard Boltzmann (1844–1906) was an Austrian physicist and philosopher and one of the founders of statistical mechanics. Ralph Kronig (1904–1995) was a German physicist. William George Penney (1909–1991) was an English mathematician and mathematical physicist. Felix Bloch (1905–1983) was a Swiss physicist and was awarded the 1952 Nobel Prize in physics. Johann Peter Gustav Lejeune Dirichlet (1805–1859) was a German mathematician with deep contributions to number theory, the theory of Fourier series and other topics in mathematical analysis. Marc-Antoine Parseval des Chênes (1755–1836) was a French mathematician. Michel Plancherel (1885–1967) was a Swiss mathematician. Takeo Matsubara (born 1921) is a Japanese theoretical physicist. Siméon Denis Poisson (1781–1840) was a French mathematician, geometer and physicist. Paul Peter Ewald (1888–1985) was a German-born US crystallographer and physicist and a pioneer of X-ray diffraction methods. Max Born (1882–1970) was a German-British physicist and mathematician. Theodore von Kármán (1881–1963) was a Hungarian-American mathematician, aerospace engineer and physicist. Léon Nicolas Brillouin (1889–1969) was a French physicist. Jean-Baptiste le Rond d’Alembert (1717–1783) was a French mathematician, mechanician, physicist, philosopher and music theorist. Hermann Ludwig Ferdinand von Helmholtz (1821–1894) was a German physician and physicist. xii Famous Scientists Mentioned in the Book

Richard Phillips Feynman (1918–1988) was an American theoretical physicist who made several fundamental contributions in physics. He received the Nobel Prize in physics in 1965. Named after Robert Brown (1773–1858) was a Scottish botanist and palaeob- otanist. Norbert Wiener (1894–1964) was an American mathematician and philosopher. Aleksandr Yakovlevich Khinchin (1894–1959) was a Soviet mathematician. Joseph Fraunhofer (1787–1826), ennobled in 1824 as Ritter von Fraunhofer, was a German optician. Pierre-Simon, marquis de Laplace (1749–1827) was a French mathematician and astronomer. Gustav Robert Kirchhoff (1824–1887) was a German physicist. Josiah Willard Gibbs (1839–1903) was an American scientist who made important theoretical contributions to physics, chemistry and mathematics. James Clerk Maxwell (1831–1879) was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the unified classical theory of electromagnetic radiation, bringing together for the first time electricity and magnetism. Pythagoras of Samos (c. 570–c. 495 BC) was an Ionian Greek philosopher and mathematician. Mikhail Vasilyevich Ostrogradsky (1801–1862) was a Ukrainian mathematician and physicist. Joseph-Louis Lagrange (1736–1813) was an Italian Enlightenment Era mathematician and astronomer who made significant contributions to the fields of analysis, number theory and both classical and celestial mechanics. Llewellyn Hilleth Thomas (1903–1992) was a British physicist and applied mathematician. Enrico Fermi (1901–1954) was an outstanding Italian physicist and the 1938 Nobel laureate in physics. Walter Kohn (born in 1923) is an Austrian-born American theoretical physicist. He was awarded, with John Pople, the Nobel Prize in chemistry in 1998 for the development of the density functional theory. Lu Jeu Sham (born April 28, 1938) is a Chinese physicist. Vladimir Aleksandrovich Fock (1898–1974) was a Soviet physicist. Douglas Rayner Hartree (1897–1958) was an English mathematician and physicist. Contents

1 Elements of Linear Algebra...... 1 1.1 Vector Spaces ...... 2 1.1.1 Introduction to Multidimensional Complex Vector Spaces ...... 2 1.1.2 AnalogyBetweenFunctionsandVectors...... 7 1.1.3 Orthogonalisation Procedure (Gram–Schmidt Method) ... 10 1.2 Matrices: Deﬁnition and Properties ...... 15 1.2.1 Introduction of a Concept...... 15 1.2.2 OperationswithMatrices...... 17 1.2.3 Inverse Matrix: An Introduction...... 23 1.2.4 LinearTransformationsanda Group...... 26 1.2.5 Orthogonal and Unitary Transformations...... 28 1.2.6 Determinantofa SquareMatrix...... 35 1.2.7 A LinearSystemofAlgebraicEquations ...... 52 1.2.8 LinearIndependenceofFunctions:Wronskian...... 56 1.2.9 CalculationoftheInverseMatrix ...... 58 1.2.10 EigenvectorsandEigenvaluesofa SquareMatrix ...... 61 1.2.11 Simultaneous Diagonalisation of Two Matrices ...... 82 1.2.12 SpectralTheoremandFunctionofa Matrix...... 85 1.2.13 QuadraticForms ...... 88 1.2.14 Extremum of a Function of n Variables...... 90 1.2.15 Traceofa Matrix...... 92 1.2.16 Tridiagonalisation of a Matrix: The Lanczos Method...... 93 1.2.17 DividingMatricesintoBlocks ...... 97 1.3 ExamplesinPhysics...... 99 1.3.1 Particleina MagneticField ...... 99 1.3.2 VibrationsinMolecules ...... 105 1.3.3 Vibrations of Atoms in an Inﬁnite Chain: A PointDefect...... 110 1.3.4 StatesofanElectronina Solid...... 116 1.3.5 TimePropagationofa WaveFunction...... 121

xiii xiv Contents

2 Complex Numbers and Functions...... 125 2.1 RepresentationofComplexNumbers ...... 125 2.2 Functionsona ComplexPlane...... 130 2.2.1 RegionsintheComplexPlaneandMapping...... 130 2.2.2 Differentiation:AnalyticFunctions...... 134 2.3 MainElementaryFunctions...... 140 2.3.1 IntegerPowerFunction...... 140 2.3.2 IntegerRootFunction...... 141 2.3.3 Exponential and Hyperbolic Functions ...... 149 2.3.4 Logarithm ...... 153 2.3.5 Trigonometric Functions...... 155 2.3.6 Inverse Trigonometric Functions ...... 158 2.3.7 GeneralPowerFunction...... 160 2.4 IntegrationintheComplexPlane...... 161 2.4.1 Deﬁnition ...... 161 2.4.2 IntegrationofAnalyticFunctions...... 164 2.5 ComplexFunctionalSeries...... 177 2.5.1 NumericalSeries...... 177 2.5.2 GeneralFunctionalSeries ...... 180 2.5.3 PowerSeries...... 182 2.5.4 TheLaurentSeries...... 186 2.5.5 Zeros and Singularities of Functions ...... 193 2.6 AnalyticContinuation...... 197 2.7 Residues...... 201 2.7.1 Deﬁnition ...... 201 2.7.2 Applications of Residues in Calculating Real AxisIntegrals ...... 205 2.8 LinearDifferentialEquations ...... 222 2.9 SelectedApplicationsinPhysics...... 229 2.9.1 DispersionRelations...... 229 2.9.2 PropagationofElectro-MagneticWavesina Material..... 232 2.9.3 Electron Tunneling in Quantum Mechanics ...... 238 2.9.4 Propagationofa QuantumState...... 246 3 Fourier Series...... 249 3.1 Trigonometric Series: An Intuitive Approach ...... 250 3.2 DirichletConditions...... 256 3.3 IntegrationandDifferentiationoftheFourierSeries...... 259 3.4 Parseval’sTheorem...... 263 3.5 Complex (Exponential) Form of the Fourier Series...... 265 3.6 ApplicationtoDifferentialEquations ...... 270 3.7 A MoreRigorousApproachtotheFourierSeries ...... 272 3.7.1 Convergence“OnAverage” ...... 272 Contents xv

3.7.2 A More Rigorous Approach to the Fourier Series:DirichletTheorem ...... 276 3.7.3 Expansion of a Function Via Orthogonal Functions ...... 280 3.8 ApplicationsofFourierSeriesinPhysics ...... 284 3.8.1 Expansions of Functions Describing Crystal Properties ... 284 3.8.2 Ewald’sFormula ...... 289 3.8.3 Born and von Karman Boundary Conditions ...... 292 3.8.4 AtomicForceMicroscopy...... 294 3.8.5 ApplicationsinQuantumMechanics ...... 297 4 Special Functions ...... 299 4.1 DiracDeltaFunction ...... 299 4.2 TheGammaFunction ...... 311 4.2.1 Deﬁnition and Main Properties...... 311 4.3 Orthogonal Polynomials ...... 318 4.3.1 Legendre Polynomials ...... 318 4.3.2 General Theory of Orthogonal Polynomials ...... 330 4.4 DifferentialEquationofGeneralisedHypergeometricType ...... 344 4.4.1 Transformationtoa StandardForm...... 344 4.4.2 SolutionsoftheStandardEquation ...... 347 4.4.3 Classical Orthogonal Polynomials ...... 352 4.5 AssociatedLegendreFunction...... 356 4.5.1 Bound Solution of the Associated Legendre Equation ..... 357 4.5.2 Orthonormality of Associated Legendre Functions ...... 361 4.5.3 LaplaceEquationinSphericalCoordinates...... 363 4.6 BesselEquation...... 369 4.6.1 BesselDifferentialEquationandItsSolutions ...... 369 4.6.2 HalfIntegerBesselFunctions...... 372 4.6.3 RecurrenceRelationsforBesselFunctions...... 373 4.6.4 Generating Function and Integral RepresentationforBesselFunctions...... 375 4.6.5 Orthogonality and Functional Series Expansion ...... 378 4.7 SelectedApplicationsinPhysics...... 382 4.7.1 Schrödinger Equation for a Harmonic Oscillator...... 382 4.7.2 SchrödingerEquationfortheHydrogenAtom...... 383 4.7.3 Stirling’sFormulaandPhaseTransitions...... 386 4.7.4 BandStructureofa Solid...... 390 4.7.5 Oscillations of a Circular Membrane ...... 394 4.7.6 MultipoleExpansionoftheElectrostaticPotential...... 397 5 Fourier Transform ...... 401 5.1 TheFourierIntegral ...... 401 5.1.1 IntuitiveApproach...... 401 5.1.2 AlternativeFormsoftheFourierIntegral...... 403 5.1.3 A MoreRigorousDerivationofFourierIntegral...... 405 xvi Contents

5.2 FourierTransform ...... 409 5.2.1 GeneralIdea...... 409 5.2.2 FourierTransformofDerivatives ...... 414 5.2.3 ConvolutionTheorem...... 416 5.2.4 Parseval’sTheorem ...... 418 5.3 ApplicationsoftheFourierTransforminPhysics ...... 419 5.3.1 Various Notations and Multiple Fourier Transform ...... 419 5.3.2 RetardedPotentials ...... 422 5.3.3 Green’sFunctionofa DifferentialEquation...... 427 5.3.4 TimeCorrelationFunctions ...... 433 5.3.5 Fraunhofer Diffraction ...... 439 6 Laplace Transform...... 445 6.1 Deﬁnition ...... 445 6.2 DetailedConsiderationoftheLaplaceTransform ...... 450 6.2.1 AnalyticityoftheLaplaceTransform...... 450 6.2.2 RelationtotheFourierTransform...... 452 6.2.3 InverseLaplaceTransform ...... 453 6.3 PropertiesoftheLaplaceTransform...... 464 6.3.1 DerivativesofOriginalsandImages...... 464 6.3.2 ShiftinImagesandOriginals ...... 467 6.3.3 IntegrationofImagesandOriginals...... 469 6.3.4 ConvolutionTheorem...... 472 6.4 SolutionofOrdinaryDifferentialEquations(ODEs)...... 473 6.5 ApplicationsinPhysics...... 479 6.5.1 Application of the Laplace Transform Method inElectronics...... 479 6.5.2 HarmonicParticlewithMemory...... 484 6.5.3 Probabilities of Hops ...... 490 6.5.4 InverseNC-AFMProblem...... 494 7 Curvilinear Coordinates ...... 499 7.1 Deﬁnition of Curvilinear Coordinates ...... 500 7.2 UnitBaseVectors...... 501 7.3 LineElementsandLineIntegral...... 509 7.4 VolumeElementandJacobianin3D ...... 512 7.5 Change of Variables in Multiple Integrals ...... 515 7.6 N-dimensionalSphere...... 520 7.7 Gradientofa ScalarField...... 523 7.8 Divergenceofa VectorField...... 525 7.9 Laplacian ...... 528 7.10 Curlofa VectorField...... 529 7.11 SomeApplicationsinPhysics...... 533 7.11.1 Partial Differential Equations of Mathematical Physics ... 533 7.11.2 ClassicalMechanicsofa Particle ...... 536 7.11.3 DistributionFunctionofa SetofParticles...... 541 Contents xvii

8 Partial Differential Equations of Mathematical Physics...... 545 8.1 GeneralConsideration...... 546 8.1.1 CharacterisationofSecondOrderPDEs...... 546 8.1.2 Initial and Boundary Conditions ...... 553 8.2 WaveEquation ...... 555 8.2.1 One-dimensionalString...... 555 8.2.2 Propagation of Sound ...... 557 8.2.3 GeneralSolutionofPDE ...... 558 8.2.4 UniquenessofSolution...... 563 8.2.5 FourierMethod...... 565 8.2.6 Forced Oscillations of the String ...... 573 8.2.7 General Boundary Problem...... 576 8.2.8 Oscillations of a Rectangular Membrane ...... 578 8.2.9 General Remarks on the Applicability oftheFourierMethod...... 582 8.3 Heat Conduction Equation ...... 585 8.3.1 UniquenessoftheSolution...... 586 8.3.2 FourierMethod...... 588 8.3.3 Stationary Boundary Conditions ...... 590 8.3.4 HeatTransportwithInternalSources...... 595 8.3.5 Solution of the General Boundary Heat Conduction Problem ...... 596 8.4 Problems Without Boundary Conditions ...... 597 8.5 ApplicationofFourierMethodtoLaplaceEquation...... 599 8.6 MethodofIntegralTransforms ...... 603 9 Calculus of Variations ...... 609 9.1 Functionsofa SingleVariable ...... 611 9.1.1 FunctionalsInvolvinga SingleFunction...... 611 9.1.2 FunctionalsInvolvingMorethanOneFunction...... 623 9.1.3 FunctionalsContainingHigherDerivatives...... 625 9.1.4 VariationwithConstraintsgivenbyZeroFunctions ...... 626 9.1.5 VariationwithConstraintsgivenbyIntegrals ...... 635 9.2 FunctionsofManyVariables...... 639 9.3 ApplicationsinPhysics...... 645 9.3.1 Mechanics...... 645 9.3.2 FunctionalDerivatives...... 654 9.3.3 Many-ElectronTheory...... 656

Index ...... 669