Mathematical Methods for Physicists: A concise introduction - Site Map

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INDEX curvilinear coordinates, 27 damped oscillations, 80 De Moivre's formula, 237 del, 22 formulas involving del, 27 delta function, Dirac, 183 Fourier integral, 183 Green's function and, 192 point source, 193 determinants, 538±547 di€erential equations, 62 ®rst order, 63 exact 67 integrating factors, 69 separable variables 63 homogeneous, 63 numerical solutions, 469 second order, constant coecients, 72 complementary functions, 74 Frobenius and Fuchs theorem, 86 particular integrals, 77 singular points, 86 solution in power series, 85 direct product matrices, 139 tensors, 50 direction angles and direction cosines, 3 divergence cartesian, 22 curvilinear, 30 cylindrical, 33 spherical polar, 35 dot product of vectors, 5 dual vectors and dual spaces, 211 eigenvalues and eigenfunctions of Sturm± Liouville equations, 340 hermitian matrices, 124 orthogonality, 129 real, 128 an operator, 217 entire functions, 247 Euler's linear equation, 83 Fourier series, 144 convergence and Dirichlet conditions, 150 di€erentiation, 157 Euler±Fourier formulas, 145 exponential form of Fourier series, 156 Gibbs phenomenon, 150 half-range Fourier series, 151 integration, 157 interval, change of, 152 orthogonality, 162 Parseval's identity, 153 vibrtating strings, 157 Fourier transform, 164 convolution theorem, 188 delta function derivation, 183 Fourier integral, 164 Fourier series (contd) Fourier sine and cosine transforms, 172 Green's function method, 192 head conduction, 179 Heisenberg's uncertainty principle, 173 Parseval's identity, 186 solution of integral equation, 421 transform of derivatives, 190 wave packets and group velocity, 174 Fredholm integral equation, 413 see also integral equations Frobenius' method see series solution of di€erential equations Frobenius±Fuch's theorem, 86 function analytic, entire, harmonic, 247 function spaces, 226 gamma function, 94 gauge transformation, 411 Gauss' law, 391 Gauss' theorem, 37 generating function, for associated Laguerre polynomials, 320 Bessel functions, 330 Hermite polynomials, 314 Laguerre polynomials, 317 Legendre polynomials, 301 Gibbs phenomenon, 150 gradient cartesian, 20 curvilinear, 29 cylindrical, 33 spherical polar, 35 Gram±Schmidt orthogonalization, 209 Green's functions, 192, construction of one dimension, 192 three dimensions, 405 delta function, 193 Green's theorem, 43 in the plane, 44 group theory, 430 conjugate clsses, 440 cosets, 439 cyclic group, 433 rotation matrix, 234, 252 special unitary group, SU(2), 232 de®nitions, 430 dihedral groups, 446 generator, 451 homomorphism, 436 irreducible representations, 442 isomorphism, 435 multiplication table, 434 Lorentz group, 454 permutation group, 438 orthogonal group SO(3) 552
INDEX group theory (contd) symmetry group, 446 unitary group, 452 unitary unimodular group SU…n† Hamilton±Jacobi equation, 364 Hamilton's principle and Lagrange equations of motion, 355 Hankel functions, 328 Hankel transforms, 385 harmonic functions, 247 Helmholtz theorem, 44 Hermite equation, 311 Hermite polynomials, 312 generating function, 314 orthogonality, 314 recurrence relations, 313 hermitian matrices, 114 orthogonal eigenvectors, 129 real eigenvalues, 128 hermitian operator, 220 completeness of eigenfunctions, 221 eigenfunctions, orthogonal, 220 eigenvalues, real, 220 Hilbert space, 230 Hilbert-Schmidt method of solution, 421 homogeneous see linear equations homomorphism, 436 indicial equation, 87 inertia, moment of, 135 in®nity see singularity, pole essential singularity integral equations, 413 Abel's equation, 426 classical harmonic oscillator, 427 di€erential equation±integral equation transformation, 419 Fourier transform solution, 421 Fredholm equation, 413 Laplace transform solution, 420 Neumann series, 416 quantum harmonic oscillator, 427 Schmidt±Hilbert method, 421 separable kernel, 414 Volterra equation, 414 integral transforms, 384 see also Fourier transform, Hankel transform, Laplace transform, Mellin transform Fourier, 164 Hankel, 385 Laplace, 372 Mellin, 385 integration, vector, 35 line integrals, 36 surface integrals, 36 interpolation, 1461 inverse operator, uniqueness of, 218 irreducible group representations, 442 Jacobian, 29 kernels of integral equations, 414 separable, 414 Kronecker delta, 6 mixed second-rank tensor, 53 Lagrangian, 355 Lagrangian multipliers, 354 Laguerre equation, 316 associated Laguerre equation, 320 Laguerre functions, 317 associated Laguerre polynomials, 320 generating function, 317 orthogonality, 319 Rodrigues' representation, 318 Laplace equation, 389 solutions of, 392 Laplace transform, 372 existence of, 373 integration of tranforms, 383 inverse transformation, 373 solution of integral equation, 420 the ®rst shifting theorem, 378 the second shifting theorem, 379 transform of derivatives, 382 Laplacian cartesian, 24 cylindrical, 34 scalar, 24 spherical polar, 35 Laurent expansion, 274 Legendre equation, 296 associated Legendre equation, 307 series solution of Legendre equation, 296 Legendre functions, 299 associated Legendre functions, 308 generating function, 301 orthogonality, 304 recurrence relations, 302 Rodrigues' formula, 299 linear combination, 204 linear independence, 204 linear operator, 212 Lorentz gauge condition, 411 Lorentz group, 454 Lorentz transformation, 455 mapping, 239 matrices, 100 anti-hermitian, 114 commutator de®nition, 100 diagonalization, 129 direct product, 139 eigenvalues and eigenvectors, 124 Hermitian, 114 nverse, 111 matrix multiplication, 103 moment of inertia, 135 orthogonal, 115 and unitary transformations, 121 553
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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 515 and 516: 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 and 564: Appendix 3 Table of * F…x† ˆp
Page 565: Index Abel's integral equation, 426
Page 569: INDEX series solution of di€erent
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