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 coumes comes from all subdisciplines of physics and engineering, the content and the level of math- ematical formalism has to be chosen very carefully. Recently the growing in- terest 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 mi mii PREFACE its subject and can be read independentIy. This makes the book very useful as a reference for scientists. We emphasize physical motivation and the mul- tidisciplinary nature of the methods discussed. The entire book contains enough material for a three-semester course meet- ing three hours a week. However, the modular structure of the book gives enough flexibility to adopt the book for several different advanced undergrad- uate and graduatelevel courses. Chapter 1 is a philosophical prelude about physics, mathematics, and mind for the interested reader. It is not a part of the curriculum for courses on mathematical methods of physics. Chapters 2-8, 12, 13 and 15-19 have been used for a tw+semester compulsory gradu- ate course meeting three hours a week. Chapters 16-20 can be used for an introductory graduate course on Green’s functions. For an upper-level un- dergraduate course on special functions, colleagues have used Chapters 1-8. Chapter 14 on fractional calculus can be expanded into a one-term elective course supported by projects given to students. Chapters 2-11 can be used in an introductory graduate course, with emphasis given to Chapters 8-11 on Stunn-Liouville theory, factorization method, coordinate transformations, general tensors, continuous groups, Lie algebras, and representations. Students are expected to be familiar with the topics generally covered dur- ing the first three years of the science and engineering undergraduate curricu- lum. These basically comprise the contents of the books Advanced Calculus by Kaplan, Introductory Complex Analysis by Brown and Churchill, and Difler- ential Equations by Ross, or the contents of books like Mathematicab Methods in Physical Sciences by Boas, Mathematical Methods: for Students of Physics and Related Fields by Hassani, and Essential Mathematical Methods for Physi- cists by Arfken and Weber. Chapters (10 and 11) on coordinates, tensors, and groups assume that the student has already seen orthogonal transformations and various coordinate systems. These are usually covered during the third year of the undergraduate physics curriculum at the level of Classical Me- chanics by Marion or Theoreticab Mechanics by Bradbury. For the sections on special relativity (in Chapter 10) we assume that the student is familiar with basic special relativity, which is usually covered during the third year of undergraduate curriculum in modern physics courses with text books like Concepts of Modern Physics by Beiser. Three very interesting chapters on the method of factorization, fractional calculus, and path integrals are included for the first time in a text book on mathematical methods. These three chapters are also extensive reviews of these subjects for beginning researchers and advanced graduate students. Summary of the Book In Chapter 1 we start with a philosophical prelude about physics, mathemat- ics, and mind. In Chapters 2-6 we present a detailed discussion of the most frequently PREFACE xviii encountered special functions in science and engineering. This is also very timely, because during the first year of graduate programs these functions are used extensively. We emphasize the fact that certain second-order par- tial differential equations are encountered in many different areas of science, thus allowing one to use similar techniques. First we approach these partial differential equations by the method of separation of variables and reduce them to a set of ordinary differential equations. They are then solved by the method of series, and the special functions are constructed by imposing appro- priate boundary conditions. Each chapter is devoted to a particular special function, where it is discussed in detail. Chapter 7 introduces hypergeometric equation and its solutions. They are very useful in parametric representations of the commonly encountered second-order differential equations and their so- lutions. Finally our discussion of special functions climaxes with Chapter 8, where a systematic treatment of their common properties is given in terms of the Sturm-Liouville theory. The subject is now approached as an eigenvalue problem for second-order linear differential operators. Chapter 9 is one of the special chapters of the book. It is a natural extension of the chapter on Sturm-Liouville theory and approaches second-order differ- ential equations of physics and engineering from the viewpoint of the theory of factorization. After a detailed analysis of the basic theory we discuss spe- cific cases. Spherical harmonics, Laguerre polynomials, Hermite polynomials, Gegenbauer polynomials, and Bessel functions are revisited and studied in detail with the factorization method. This method is not only an interesting approach to solving Sturm-Liouville systems, but also has deep connections with the symmetries of the system. Chapter 10 presents an extensive treatment of coordinates, their transfor- mations, and tensors. We start with the Cartesian coordinates, their trans- formations, and Cartesian tensors. The discussion is then extended to general coordinate transformations and general tensors. We also discuss Minkowski spacetime, coordinate transformations in spacetime, and four-tensors in de- tail. We also write Maxwell’s equations and Newton’s dynamical theory in covariant form and discuss their transformation properties in spacetime. In Chapter 11 we discuss continuous groups, Lie algebras, and group rep- resentations. Applications to the rotation group, special unitary group, and homogeneous Lorentz group are discussed in detail. An advanced treatment of spherical harmonics is given in terms of the rotation group and its repre sentations. We also discuss symmetry of differential equations and extension (prolongation) of generators. Chapters 12 and 13 deal with complex analysis. We discuss the theory of analytic functions, mappings, and conformal and Schwarz-Christoffel trans- formations with interesting examples like the fringe effects of a parallel plate capacitor and fluid flow around an obstacle. We also discuss complex inte- grals, series, and analytic continuation along with the methods of evaluating some definite integrals. Chapter 14 introduces the basics of fractional calculus. After introducing xxiv PREFACE the experimental motivation for why we need fractional derivatives and inte- grals, we give a unified representation of the derivative and integral and extend it to fractional orders. Equivalency of different definitions, examples, prop erties, and techniques with fractional derivatives are discussed. We conclude with examples from Brownian motion and the Fokker-Planck equation. This is an emerging field with enormous potential and with applications to physics, chemistry, biology, engineering, and finance. For beginning researchers and instructors who want to add something new and interesting to their course, this self-contained chapter is an excellent place to start. Chapter 15 contains a comprehensive discussion of infinite series: tests of convergence, properties, power series, and uniform convergence along with the methods of evaluating sums of infinite series. An interesting section on divergent series in physics is added with a discussion of the Casimir effect. Chapter 16 treats integral transforms. We start with the general defini- tion, and then the two most commonly used integral transforms, Fourier and Laplace transforms, are discussed in detail with their various applications and techniques. Chapter 17 is on variational analysis. Cases with different numbers of de- pendent and independent variables are discussed. Problems with constraints, variational techniques in eigenvalue problems, and the Rayleigh-Ritz method are among other interesting topics covered. In Chapter 18 we introduce integral equations. We start with their classifi- cation and their relation to differential equations and vice versa. We continue with the methods of solving integral equations and conclude with the eigen- value problem for integral operators, that is, the Hilbert-Schmidt theory. In Chapter 19 (and 20) we present Green’s functions, and this is the second climax of this book, where everything discussed so far is used and their con- nections seen. We start with the timeindependent Green’s functions in one dimension and continue with three-dimensional Green’s functions. We discuss their applications to electromagnetic theory and the Schrijdinger equation. Next we discuss first-order time-dependent Green’s functions with applica- tions to diffusion problems and the timedependent Schrodinger equation. We introduce the propagator interpretation and the compounding of propagators. We conclude this section with second-order time-dependent Green’s functions, and their application to the wave equation and discuss advanced and retarded soh tions. Chapter 20 is an extensive discussion of path integrals and their relation to Green’s functions. During the past decade or so path integrals have found wide range of applications among many different fields ranging from physics to finance. We start with the Brownian motion, which is considered a pro- totype of many different processes in physics, chemistry, biology, finance etc. We discuss the Wiener path integral approach to Brownian motion. After the Feynman-Kac formula is introduced, the perturbative solution of the Bloch equation is given. Next an interpretation of V(z) in the Bloch equation is given, and we continue with the methods of evaluating path integrals. We PREFACE xxv also discuss the Feynman path integral formulation of quantum mechanics along with the phase space approach to Feynman path integrals. Story of the Book Since 1989, I have been teaching the graduate level ‘Methods of Mathematical Physics I & 11’ courses at the Middle East Technical University in Ankara. Chapters 2-8 with 12 and 13 have been used for the first part and Chapters 15-19 for the second part of this course, which meets three hours a week. Whenever possible I prefer to introduce mathematical techniques through physical applications. Examples are often used to extend discussions of spe- cific 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 gen- eral theory, as in Chapter 8 (and 9) on the Sturm-Liouville theory, or with a part that utilizes the gains of the previous chapters, as in Chapter 19 (and 20) on Green’s functions. Chapter 9 is on factorization method, which is a natural extension of our discussion on the Sturm-Liouville theory. It also presents a different and advanced treatment of special functions. Similarly, Chapter 20 on path integrals is a natural extension of our chapter on Green’s functions. Chapters 10 and 11 on coordinates, tensors, and continuous groups have been located after Chapter 9 on the Sturm-Liouville theory and the fac- torization method. Chapters 12 and 13 are on complex techniques, and they are self-contained. Chapter 14 on fractional calculus can either be integrated into the curriculum of the mathematical methods of physics courses or used independently. 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 discussions. Later, when they go through the derivations, technical details become understood, alternate approaches appear, and some of the questions are 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. 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. Sometimes I indicated this in the text itself, but a complete list is given at the back. Readers of this book will hopefully be well prepared for advanced graduate studies in many areas of physics. In particular, as we use the same terminol- ogy and style, they should be ready for full-term graduate courses based on the books: The Fractional Calculus by Oldham and Spanier and Path Inte- xxvi PREFACE gmls in Physics, Volumes I and 11 by Chaichian and Demichev, or they could jump into the advanced sections of these books, which have become standard references in their fields. I recommend that students familiarize themselves with the existing litera- ture. Except for an isolated number of instances I have avoided giving refer- ences within the text. The references at the end should be a good first step in the process of meeting the literature. In addition to the references at the back, there are also three websites that are invaluable to students and researchers: For original research, http://lanl.arxiv.org/ and the two online encyclope- dias: http://en.wikipedia.org and http://scienceworld.wolfram.com/ are very useful. For our chapters on special functions these online encyclopedias are extremely helpful with graphs and additional information. A precursor of this book (Chapters 1-8, 12, 13, and 1519) was published in Turkish in 2000. With the addition of two new chapters on fractional calculus and path integrals, the revised and expanded version appeared in 2004 as 440 pages and became a widely used text among the Turkish universities. The pos- itive feedback from the Turkish versions helped me to prepare this book with a minimum number of errors and glitches. For news and communications about the book we will use the website http://www.physics.metu.edu.tr/- bayin, which will also contain some relevant links of interest to readers. S. BAYIN OD TU Ankam/TURKE Y April 2006 Acknowledgments I would like to pay 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. I have benefitted from them greatly. 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 1 and 9 with 14 and 20. I also thank Prof. N. K. Pak for many interesting and stimulating discussions, encouragement, and critical reading of the chapter on path integrals. I thank Wiley for the support by a grant during the preparation of the camera ready copy. My special thanks go to my editors at Wiley, Steve Quigley, Susanne Steitz, and Danielle Lacourciere for sharing my excitement and their utmost care in bringing this book into existence. I finally thank my wife, Adalet, and daughter, Sumru, for their endless support during the long and strenuous period of writing, which spanned over several years. 3.S.B. xxvii This Page Intentionally Left Blank NATURE and MATHEMATICS The most incomprehensible thing about this universe is that it is comprehensible - Albert Einstein When man first opens his eyes into this universe, he encounters an endless variety of events and shivers as he wonders how he will ever survive in this enormously complex system. However, as he contemplates he begins to realize that the universe is not hostile and there is some order among all this diversity. As he wanders around, he inadvertently kicks stones on his path. As the stones tumble away, he notices that the smaller stones not only do not hurt his feet, but also go further. Of course, he quickly learns to avoid the bigger ones. The sun, to which he did not pay too much attention at first, slowly begins to disappear; eventually leaving him in cold and dark. At first this scares him a lot. However, what a joy it must be to witness the sun slowly reappearing in the horizon. As he continues to explore, he realizes that the order in this universe is also dependable. Small stones, which did not hurt him, do not hurt him another day in another place. Even though the sun eventually disappears, leaving him in cold and dark, he is now confident that it will reappear. In time he learns to live in communities and develops languages to communicate with his fellow human beings. Eventually the quality and the number of observations he makes increase. In fact, he even begins to undertake projects that require careful recording and interpretation of data that span over several generations. As in Stonehenge he even builds an agricultural computer to find the crop times. A similar version of this story is actually repeated with every newborn. 1 2 NATURE AND MATHEMATICS For man to understand nature and his place in it has always been an in- stinctive desire. Along this endeavour he eventually realizes that the everyday language developed to communicate with his fellow human beings is not suf- ficient. For further understanding of the law and order in the universe, a new language, richer and more in tune with the inner logic of the universe, is needed. At this point physics and mathematics begin to get acquainted. With the discovery of coordinate systems, which is one of the greatest con- structions of the free human mind, foundations of this relation become ready. Once a coordinate system is defined, it is possible to reduce all the events in the universe to numbers. Physical processes and the law and order that ex- ists among these events can now be searched among these numbers and could be expressed in terms of mathematical constructs much more efficiently and economically. From the motion of a stone to the motions of planets and stars, it can now be understood and expressed in terms of the dynamical theory of Newton: 87 T=m- dt2 and his law of gravitation Newton’s theory is full of the success stories that very few theories will ever have for years to come. Among the most dramatic is the discovery of Nep tune. At the time small deviations from the calculated orbit of Uranus were observed. At first the neighboring planets, Saturn and Jupiter, were thought to be the cause. However, even after the effects of these planets were sub- tracted, a small unexplained difference remained. Some scientists questioned even the validity of Newton’s theory. However, astronomers, putting their trust in Newton’s theory, postulated the existence of another planet as the source of these deviations. From the amount of the deviations they calculated the orbit and the mass of this proposed planet. They even gave a name to it: Neptune. Now the time had come to observe this planet. When the telescopes were turned into the calculated coordinates: Hello! Neptune was there. In the nineteenth century, when Newton’s theory was joined by Maxwell’s the- ory of electromagnetism, there was a time when even the greatest minds like Bertrand Russell began to think that physics might have come to an end, that is, the existing laws of nature could in principle explain all physical phenom- ena. Actually, neither Newton’s equations nor Maxwell’s equations are laws in the strict sense. They are based on some assumptions. Thus it is proba- bly more appropriate to call them theories or models. We frequently make assumptions in science. Sometimes in order to concentrate on a special but frequently encountered case, we keep some of the parameters constant to avoid [...]... 1 (1 - t (2x - t ) ) 2 I=O 22 ' ( l ! ) 2 (22 -t) 1 (2. 70) Using the binomial formula again, we expand the factor (2x - t )1 t o write 2 1=0 (2I)! (-1 )21 t l 22 1 c 1 (q2 k=O k! ( I I! k ) ! ( - (2. 71) 2 2 y (-ty (2. 72) (2. 73) 1=0 k=O LEGENDRE POLYNOMIALS 21 We now rearrange the double sum by the substitutions k i n and 2 4 2 - n (2. 74) to write n=O C2 1 ( 2 - n)!n!1 - 2n)! ( (2. 75) Comparing this with... l in the generating function Equation (2. 65) we find (2. 85) Expanding the left-hand side by using the binomial formula and comparing equal powers o f t , we obtain 9 (1) = 1 and 8 (-1) = (-1) 1 (2. 86) Similarly, we write x = 0 in the generating function t o get 00 (2. 87) (2. 88) This leads us t o the special values: and (-$ p21 (O) = (21 )! 22 1 (1! )2 (2. 90) 23 LEGENDRE POLYNOMIALS 2. 3.5 Special Integrals... be solved for 0 (Q )and sin2Q - d 0 (0) 2 dd2 dO ( Q ) +cosQsinQd8 + [Xsin2Q-m2] O(8) = 0 (2. 15) (4) as (2. 16) 12 LEGENDRE EQUATION AND POLYNOMIALS and (2. 17) In summary, using the method of separation of variables we have reduced the partial differential Equation (2. 9) to three ordinary differential Equations, (2. 12) , (2. 16), and (2. 17) During this process two constant parameters, X and m,called the... ( 1 - x 2 ) x ] +2( 1+1)fi(z) dw dx=O (2. 101) Using integration by parts this can be written as 1' [ -1 (z2- 1) -dx + 1 ( 1 + 9 dx 1)fit ( z ) f i (z) dx = 0 (2. 1 02) Interchanging 1 and 2' in Equation (2. 1 02) and subtracting the result from Equation (2. 1 02) we get [2 (1 / + 1) - 2' (2' + l)] 1 -1 f i f (z) 4 (x)CliE = 0 (2. 103) For 1 # 2' this equation gives (2. 104) LEGENDRE POLYNOMIALS 25 and for 1... the 21 -fold derivative of (x2 - 1)l as (21 )! Thus Equation (2. 108) becomes (2. 110) 1 We now write (1 - x2) as (1 - 2) = (1 - x2) (1 - x 1 y = (1 - x y + -x d 21 dx (1 - ")' (2. 111) to obtain N[ = (21 - 1) (21 - 'i! l N1-1+ - l x d [(1 - x'),'] 21 22 1 ( l ! ) (2. 1 12) (21 - 1) "-1 21 (2. 113) This gives Ni = ~ 1 21 - -Nl or (2. 114) 26 LEGENDRE EQUATION AND POLYNOMIALS which means that the value of (21 ... (2. 61) to write Equation (2. 60) as c 1 d' a (x)= 2" ! dx' We now use the formula n=O l!(-l)n x21-2n (2. 62) n! (1 - n)! (2. 63) to obtain (2. 64) thus proving the equivalence of Equations (2. 60) and (2. 59) 2. 3 .2 Generating Function Another way to define the Legendre polynomials is by using a generating function, T (x,) , which is defined as t T (z,t) = 1 cfi( x ) t ' , 00 dl - 2xt + t 2 = I1 < 1 t (2. 65)... equation Assuming a0 # 0, the two roots of the indicial equation give the values of a a s a = 0 and a = 1 (2. 32) The remaining Equations (2. 30) and (2. 31) give us the recursion relation among the remaining coefficients Starting with a = 1 we obtain ak +2 =ak + + ( k 1) ( k 2) - x , k = 0 , 1 , 2 ( k 2) ( k 3) + + (2. 33) = 0, (2. 34) For a = 1 Equation (2. 30) implies a1 hence all the remaining nonzero coefficients... ~(z) and integrate over z and use the J orthogonality relation to get B i 2 e ( a , z )E j (z) dx = - _ a"+' (21 + 1) ' (2. 129 ) 28 LEGENDRE EQUATiON AND POLYNOMIALS Bl = (21 + 1) al+l 2 (2. 131) For the even values of 1 the expansion coefficients are zero For the odd values of 1 we use the result Equation (2. 95) to write &s+l avo), 3, p2s (O) U2S +2( 2 (2s +2) = (4s + s = 0,1 ,2, (2. 1 32) Substituting... t (22 - t ) ] a (2. 66) and use the binomial expansion (2. 67) 20 LEGENDRE EQUATION AND POLYNOMIALS We derive the useful relation: < I - (- 1)l [ (;) (f + (f + 1) 2) [(-;-1) [(;) (f + = (-1)' = (-1) = (-1) [ 1) (-a -1) (-f ] - 1 - 1) (-f-I-l) ] (a + 2) (; + I - l)] 1-3-5-.- (21 -1) 21 1 (21 )! 22 11! ' (2. 68) t o write Equation (2. 67) as (2. 69) We use this in Equation (2. 66) to write =c (2I)! (-1 )21 ... constant Now Equation (2. 11) reduces to the following two equations: (2. 12) and sin 8 dQ [sin0 dl9 (” ) ’ + XY (Q, 4 ) = 0 (2. 13) Equation (2. 12) for R(r) is now an ordinary differential equation We also as separate the Q and the (b variables in Y (Q,4) and call the new separation constant m2 ,and write 1 d 2 @ ( 4 ) = m2 d [sin@%] + X s i n 2 Q = sin8 0 ( 8 ) dd @(4) We now obtain the differential . 0 (Q) and (4) as dO (Q) d8 +cosQsinQ- + [Xsin2Q-m2] O(8) = 0 sin2 Q- (2. 16) d20 (0) dd2 12 LEGENDRE EQUATION AND POLYNOMIALS and (2. 17) In summary, using the method of. r2dr dr r2 sin Q dQ ae (2. 9) 1 82 r2 sin2 Q 84’ + R (r) Y (Q, 4) + k2 (r) R (r) Y (8,4) = 0. After multiplying the above equation by r2 (2. lo) and collecting. the Q and the (b variables in Y (Q,4) as and call the new separation constant m2, and write (2. 15) sin8 d [sin@%] +Xsin2Q= 1 d2@(4) =m 2 . 0 (8) dd @(4) We now obtain the