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Further reading Anton, Howard, Elementary Linear Algebra, 3rd ed., John Wiley, New York, 1982. Arfken, G. B., Weber, H. J., Mathematical Methods for Physicists, 4th ed., Academic Press, New York, 1995. Boas, Mary L., Mathematical Methods in the Physical Sciences, 2nd ed., John Wiley, New York, 1983. Butkov, Eugene, Mathematical Physics, Addison-Wesley, Reading (MA), 1968. Byon, F. W., Fuller, R. W., Mathematics of Classical and Quantum Physics, Addison- Wesley, Reading (MA), 1968. Churchill, R. V., Brown, J. W., Verhey, R. F., Complex Variables & Applications, 3rd ed., McGraw-Hill, New York, 1976. Harper, Charles, Introduction to Mathematical Physics, Prentice Hall, Englewood Cli€s, NJ, 1976. Kreyszig, E., Advanced Engineering Mathematics, 3rd ed., John Wiley, New York, 1972. Joshi, A. W., Matrices and Tensor in Physics, John Wiley, New York, 1975. Joshi, A. W., Elements of Group Theory for Physicists, John Wiley, New York, 1982. Lass, Harry, Vector and Tensor Analysis, McGraw-Hill, New York, 1950. Margenus, Henry, Murphy, George M., The Mathematics of Physics and Chemistry, D. Van Nostrand, New York, 1956. Mathews, Fon, Walker, R. L., Mathematical Methods of Physics, W. A. Benjamin, New York, 1965. Spiegel, M. R., Advanced Mathematics for Engineers and Scientists, Schaum's Outline Series, McGraw-Hill, New York, 1971. Spiegel, M. R., Theory and Problems of Vector Analysis, Schaum's Outline Series, McGraw-Hill, New York, 1959. Wallace, P. R., Mathematical Analysis of Physical Problems, Dover, New York, 1984. Wong, Chun Wa, Introduction to Mathematical Physics, Methods and Concepts, Oxford, New York, 1991. Wylie, C., Advanced Engineering Mathematics, 2nd ed., McGraw-Hill, New York, 1960. 549
Index Abel's integral equation, 426 Abelian group, 431 adjoint operator, 212 analytic functions, 243 Cauchy integral formula and, 244 Cauchy integral theorem and, 257 angular momentum operator, 18 Argand diagram, 234 associated Laguerre equation, polynomials see Laguerre equation; Laguerre functions associated Legendre equation, functions see Legendre equation; Legendre functions associated tensors, 53 auxiliary (characteristic) equation, 75 axial vector, 8 Bernoulli equation, 72 Bessel equation, 321 series solution, 322 Bessel functions, 323 approximations, 335 ®rst kind J n …x†, 324 generating function, 330 hanging ¯exible chain, 328 Hankel functions, 328 integral representation, 331 orthogonality, 336 recurrence formulas, 332 second kind Y n …x† see Neumann functions spherical, 338 beta function, 95 branch line (branch cut), 241 branch points, 241 calculus of variations, 347±371 brachistochrone problem, 350 canonical equations of motion, 361 constraints, 353 Euler±Lagrange equation, 348 Hamilton's principle, 361 calculus of variations (contd) Hamilton±Jacobi equation, 364 Lagrangian equations of motion, 355 Lagrangian multipliers, 353 modi®ed Hamilton's principle, 364 Rayleigh±Ritz method, 359 cartesian coordinates, 3 Cauchy principal value, 289 Cauchy±Riemann conditions, 244 Cauchy's integral formula, 260 Cauchy's integral theorem, 257 Cayley±Hamilton theorem, 134 change of basis, 224 coordinate system, 11 interval, 152 characteristic equation, 125 Christo€el symbol, 54 commutator, 107 complex numbers, 233 basic operations, 234 polar form, 234 roots, 237 connected, simply or multiply, 257 contour integrals, 255 contraction of tensors, 50 contravariant tensor, 49 convolution theorem Fourier transforms, 188 coordinate system, see speci®c coordinate system coset, 439 covariant di€erentiation, 55 covariant tensor, 49 cross product of vectors see vector product of vectors crossing conditions, 441 curl cartesian, 24 curvilinear, 32 cylindrical, 34 spherical polar, 35 551
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Mathematical Methods for Physicists
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PUBLISHED BY CAMBRIDGE UNIVERSITY P
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Contents Preface xv 1 Vector and te
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CONTENTS 3 Matrix algebra 100 De®n
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CONTENTS The adjoint operators 219
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CONTENTS Sturm±Liouville systems 3
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CONTENTS The Runge±Kutta method 47
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Preface This book evolved from a se
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VECTOR AND TENSOR ANALYSIS Figure 1
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VECTOR AND TENSOR ANALYSIS Vector a
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VECTOR AND TENSOR ANALYSIS We can g
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VECTOR AND TENSOR ANALYSIS Using th
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VECTOR AND TENSOR ANALYSIS The trip
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VECTOR AND TENSOR ANALYSIS will be
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VECTOR AND TENSOR ANALYSIS orthonor
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VECTOR AND TENSOR ANALYSIS A(u) is
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VECTOR AND TENSOR ANALYSIS a circle
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VECTOR AND TENSOR ANALYSIS The vect
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VECTOR AND TENSOR ANALYSIS operate
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VECTOR AND TENSOR ANALYSIS We ®rst
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VECTOR AND TENSOR ANALYSIS system w
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VECTOR AND TENSOR ANALYSIS we have
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VECTOR AND TENSOR ANALYSIS with sim
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VECTOR AND TENSOR ANALYSIS where A
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VECTOR AND TENSOR ANALYSIS Z P2 P 1
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VECTOR AND TENSOR ANALYSIS name is
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VECTOR AND TENSOR ANALYSIS while th
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VECTOR AND TENSOR ANALYSIS Next we
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VECTOR AND TENSOR ANALYSIS This imp
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VECTOR AND TENSOR ANALYSIS Proof:
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VECTOR AND TENSOR ANALYSIS Here we
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VECTOR AND TENSOR ANALYSIS (2) Addi
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VECTOR AND TENSOR ANALYSIS A genera
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VECTOR AND TENSOR ANALYSIS two poin
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VECTOR AND TENSOR ANALYSIS which sh
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2 Ordinary di€erential equations
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ORDINARY DIFFERENTIAL EQUATIONS The
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ORDINARY DIFFERENTIAL EQUATIONS Sol
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ORDINARY DIFFERENTIAL EQUATIONS It
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ORDINARY DIFFERENTIAL EQUATIONS whe
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ORDINARY DIFFERENTIAL EQUATIONS Gen
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ORDINARY DIFFERENTIAL EQUATIONS and
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ORDINARY DIFFERENTIAL EQUATIONS (1)
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ORDINARY DIFFERENTIAL EQUATIONS Now
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ORDINARY DIFFERENTIAL EQUATIONS Obv
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ORDINARY DIFFERENTIAL EQUATIONS Sol
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ORDINARY DIFFERENTIAL EQUATIONS Ord
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ORDINARY DIFFERENTIAL EQUATIONS Cas
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ORDINARY DIFFERENTIAL EQUATIONS Int
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ORDINARY DIFFERENTIAL EQUATIONS of
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ORDINARY DIFFERENTIAL EQUATIONS and
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ORDINARY DIFFERENTIAL EQUATIONS Fig
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3 Matrix algebra As vector methods
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MATRIX ALGEBRA Four basic algebra o
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MATRIX ALGEBRA then ~A ~B ˆ ˆ 2
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MATRIX ALGEBRA Figure 3.1. Coordina
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MATRIX ALGEBRA in®nite sum of matr
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MATRIX ALGEBRA The product of two s
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MATRIX ALGEBRA ~B ~ A ˆ ~I. Multip
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MATRIX ALGEBRA but so that ~A 1 …
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MATRIX ALGEBRA But if ~ A is an in
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MATRIX ALGEBRA Taking the dot produ
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MATRIX ALGEBRA change the handednes
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MATRIX ALGEBRA preserves the norm o
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MATRIX ALGEBRA Then ~A 0 R 0 ˆ…
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MATRIX ALGEBRA Once the eigenvalues
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MATRIX ALGEBRA Example 3.15 Show th
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0 1 x 11 x 1i x 1n x 21 x 2i x
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MATRIX ALGEBRA Since 0 1 1 B @ 1 0
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MATRIX ALGEBRA We now proceed to pr
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MATRIX ALGEBRA distance between any
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MATRIX ALGEBRA This matrix equation
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~A ~B ˆ a11 B ~ ! a 12 ~B ˆ a 21
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MATRIX ALGEBRA 3.18 If A ~ ~B ˆ 0,
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4 Fourier series and integrals Four
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FOURIER SERIES AND INTEGRALS Fourie
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FOURIER SERIES AND INTEGRALS Soluti
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FOURIER SERIES AND INTEGRALS Figure
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FOURIER SERIES AND INTEGRALS but b
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FOURIER SERIES AND INTEGRALS Parsev
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FOURIER SERIES AND INTEGRALS the A
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FOURIER SERIES AND INTEGRALS Orthog
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FOURIER SERIES AND INTEGRALS Substi
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FOURIER SERIES AND INTEGRALS and th
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FOURIER SERIES AND INTEGRALS We sho
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FOURIER SERIES AND INTEGRALS and th
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FOURIER SERIES AND INTEGRALS contin
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FOURIER SERIES AND INTEGRALS amount
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FOURIER SERIES AND INTEGRALS Using
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FOURIER SERIES AND INTEGRALS It is
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FOURIER SERIES AND INTEGRALS Parsev
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FOURIER SERIES AND INTEGRALS This i
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FOURIER SERIES AND INTEGRALS Soluti
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FOURIER SERIES AND INTEGRALS The de
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FOURIER SERIES AND INTEGRALS By the
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FOURIER SERIES AND INTEGRALS 4.10 F
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FOURIER SERIES AND INTEGRALS 4.22 V
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LINEAR VECTOR SPACES Figure 5.1. A
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LINEAR VECTOR SPACES We now proceed
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LINEAR VECTOR SPACES Linear combina
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LINEAR VECTOR SPACES is a basis for
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LINEAR VECTOR SPACES then Eq. (5.10
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LINEAR VECTOR SPACES Step 3. Clearl
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LINEAR VECTOR SPACES then hujvi ˆh
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LINEAR VECTOR SPACES Eq. (5.16) det
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LINEAR VECTOR SPACES In general A ~
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LINEAR VECTOR SPACES The inverse of
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LINEAR VECTOR SPACES It is also eas
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LINEAR VECTOR SPACES or U ~ ‡ ˆ
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LINEAR VECTOR SPACES Change of basi
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LINEAR VECTOR SPACES Multiplying Eq
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LINEAR VECTOR SPACES For a vibratin
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LINEAR VECTOR SPACES the integratio
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LINEAR VECTOR SPACES (b) T…a† i
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FUNCTIONS OF A COMPLEX VARIABLE and
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FUNCTIONS OF A COMPLEX VARIABLE We
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FUNCTIONS OF A COMPLEX VARIABLE Fig
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FUNCTIONS OF A COMPLEX VARIABLE ver
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FUNCTIONS OF A COMPLEX VARIABLE z 0
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FUNCTIONS OF A COMPLEX VARIABLE If
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FUNCTIONS OF A COMPLEX VARIABLE How
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FUNCTIONS OF A COMPLEX VARIABLE Sim
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FUNCTIONS OF A COMPLEX VARIABLE Di
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FUNCTIONS OF A COMPLEX VARIABLE fun
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FUNCTIONS OF A COMPLEX VARIABLE Sin
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FUNCTIONS OF A COMPLEX VARIABLE pro
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FUNCTIONS OF A COMPLEX VARIABLE Alt
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FUNCTIONS OF A COMPLEX VARIABLE Exa
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FUNCTIONS OF A COMPLEX VARIABLE Exa
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FUNCTIONS OF A COMPLEX VARIABLE For
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FUNCTIONS OF A COMPLEX VARIABLE Sol
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FUNCTIONS OF A COMPLEX VARIABLE Rat
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FUNCTIONS OF A COMPLEX VARIABLE Not
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FUNCTIONS OF A COMPLEX VARIABLE The
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FUNCTIONS OF A COMPLEX VARIABLE (b)
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FUNCTIONS OF A COMPLEX VARIABLE to
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FUNCTIONS OF A COMPLEX VARIABLE Sol
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FUNCTIONS OF A COMPLEX VARIABLE the
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FUNCTIONS OF A COMPLEX VARIABLE Thu
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FUNCTIONS OF A COMPLEX VARIABLE If
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FUNCTIONS OF A COMPLEX VARIABLE the
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FUNCTIONS OF A COMPLEX VARIABLE onl
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FUNCTIONS OF A COMPLEX VARIABLE To
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FUNCTIONS OF A COMPLEX VARIABLE Tak
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FUNCTIONS OF A COMPLEX VARIABLE 6.2
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7 Special functions of mathematical
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SPECIAL FUNCTIONS OF MATHEMATICAL P
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THE CALCULUS OF VARIATIONS Figure 8
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THE CALCULUS OF VARIATIONS This is
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THE CALCULUS OF VARIATIONS p Lettin
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THE CALCULUS OF VARIATIONS provided
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THE CALCULUS OF VARIATIONS and Eq.
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THE CALCULUS OF VARIATIONS Example
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THE CALCULUS OF VARIATIONS must van
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THE CALCULUS OF VARIATIONS mechanic
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THE CALCULUS OF VARIATIONS Solution
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THE CALCULUS OF VARIATIONS The acti
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THE CALCULUS OF VARIATIONS is to ®
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THE CALCULUS OF VARIATIONS y…"; x
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9 The Laplace transformation The La
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THE LAPLACE TRANSFORMATION This est
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THE LAPLACE TRANSFORMATION or L‰x
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THE LAPLACE TRANSFORMATION Recall L
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THE LAPLACE TRANSFORMATION Let u ˆ
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THE LAPLACE TRANSFORMATION But f
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THE LAPLACE TRANSFORMATION but and
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THE LAPLACE TRANSFORMATION 9.6 Find
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PARTIAL DIFFERENTIAL EQUATIONS prob
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PARTIAL DIFFERENTIAL EQUATIONS If
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PARTIAL DIFFERENTIAL EQUATIONS In t
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PARTIAL DIFFERENTIAL EQUATIONS The
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PARTIAL DIFFERENTIAL EQUATIONS We a
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PARTIAL DIFFERENTIAL EQUATIONS Let
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PARTIAL DIFFERENTIAL EQUATIONS brin
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PARTIAL DIFFERENTIAL EQUATIONS We n
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PARTIAL DIFFERENTIAL EQUATIONS whic
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PARTIAL DIFFERENTIAL EQUATIONS to t
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PARTIAL DIFFERENTIAL EQUATIONS Subs
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PARTIAL DIFFERENTIAL EQUATIONS Now
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PARTIAL DIFFERENTIAL EQUATIONS Figu
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SIMPLE LINEAR INTEGRAL EQUATIONS f
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SIMPLE LINEAR INTEGRAL EQUATIONS Th
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SIMPLE LINEAR INTEGRAL EQUATIONS wh
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SIMPLE LINEAR INTEGRAL EQUATIONS in
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SIMPLE LINEAR INTEGRAL EQUATIONS A
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SIMPLE LINEAR INTEGRAL EQUATIONS no
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SIMPLE LINEAR INTEGRAL EQUATIONS Th
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SIMPLE LINEAR INTEGRAL EQUATIONS wh
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12 Elements of group theory Group t
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ELEMENTS OF GROUP THEORY The same s
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ELEMENTS OF GROUP THEORY Figure 12.
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ELEMENTS OF GROUP THEORY Table 12.3
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ELEMENTS OF GROUP THEORY Obviously
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ELEMENTS OF GROUP THEORY This can b
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ELEMENTS OF GROUP THEORY Group repr
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ELEMENTS OF GROUP THEORY (1) A matr
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ELEMENTS OF GROUP THEORY Equation (
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ELEMENTS OF GROUP THEORY Table 12.6
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ELEMENTS OF GROUP THEORY met. In qu
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ELEMENTS OF GROUP THEORY 0 1 1 0 0
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ELEMENTS OF GROUP THEORY Figure 12.
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ELEMENTS OF GROUP THEORY We can exp
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ELEMENTS OF GROUP THEORY Next, we a
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NUMERICAL METHODS is rather tedious
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NUMERICAL METHODS Figure 13.2. cour
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NUMERICAL METHODS and take g…x†
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NUMERICAL METHODS Solution: Here f
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NUMERICAL METHODS where y i ˆ f
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NUMERICAL METHODS The ®rst-order o
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NUMERICAL METHODS approximate value
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NUMERICAL METHODS Runge±Kutta meth
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NUMERICAL METHODS Equations of high
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NUMERICAL METHODS We now illustrate
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NUMERICAL METHODS 13.14. Find to th
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INTRODUCTION TO PROBABILITY THEORY
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Page 513 and 514: INTRODUCTION TO PROBABILITY THEORY
Page 521 and 522: Appendix 1 Preliminaries (review of
Page 523 and 524: APPENDIX 1 PRELIMINARIES Problem A1
Page 527 and 528: APPENDIX 1 PRELIMINARIES Example A1
Page 529 and 530: APPENDIX 1 PRELIMINARIES Since the
Page 531 and 532: APPENDIX 1 PRELIMINARIES Figure A1.
Page 533 and 534: APPENDIX 1 PRELIMINARIES How do we
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Page 539 and 540: APPENDIX 1 PRELIMINARIES Theorems o
Page 541 and 542: APPENDIX 1 PRELIMINARIES This is th
Page 543 and 544: APPENDIX 1 PRELIMINARIES (d) Find t
Page 545 and 546: APPENDIX 1 PRELIMINARIES Divide the
Page 547 and 548: APPENDIX 1 PRELIMINARIES Example A1
Page 549 and 550: APPENDIX 1 PRELIMINARIES Di€erent
Page 551 and 552: APPENDIX 1 PRELIMINARIES derivative
Page 553 and 554: Appendix 2 Determinants The determi
Page 555 and 556: APPENDIX 2 DETERMINANTS where a 11
Page 557 and 558: APPENDIX 2 DETERMINANTS D ˆ a i1 C
Page 559 and 560: APPENDIX 2 DETERMINANTS Example A2.
Page 561 and 562: APPENDIX 2 DETERMINANTS Proof: Expa
Page 563: Appendix 3 Table of * F…x† ˆp
Page 567 and 568: INDEX group theory (contd) symmetry
Page 569: INDEX series solution of di€erent
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