Essential mathematical methods for physicists, hans j weber, george b arfken

www.elsolucionario.net www.elsolucionario.net Vector Identities A = Ax xˆ + Ayyˆ + Azzˆ , A2 = Ax2 + A2y + A2z , Ay Az By Bz Ax Ay A · (B × C) = Bx Cx By A×B = Cy xˆ − Ax Az Bx Bz yˆ + Az Ay Bz = C x By Cz A × (B × C) = B A · C − C A · B, Az Bz A · B = Ax Bx + Ay By + Az Bz Ax Ay Bx By − Cy zˆ Ax Az Bx Bz + Cz Ax Ay Bx By εijk ε pqk = δip δ jq − δiq δ jp k Vector Calculus F = −∇V (r) = − r dV dV = −ˆr , r dr dr ∇ · (r f (r)) = f (r) + r df , dr ∇ · (rr n−1 ) = (n + 2)r n−1 ∇(A · B) = (A · ∇)B + (B · ∇)A + A × (∇ × B) + B × (∇ × A) ∇ · (SA) = ∇S · A + S∇ · A, ∇ · (∇ × A) = 0, ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) ∇ × (SA) = ∇S × A + S∇ × A, ∇ × (r f (r)) = 0, ∇×r = ∇ × (A × B) = A ∇ · B − B ∇ · A + (B · ∇)A − (A · ∇)B, ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A ∇ · B d3 r = V B · da, (∇ × A) · da = (Gauss), S (φ∇2 ψ − ψ∇2 φ)d3r = V ∇2 = −4π δ(r), r A · dl, (Stokes) (φ∇ψ − ψ∇φ) · da, (Green) S S δ(ax) = δ(x), |a| δ(t − x) = 2π δ( f (x)) = δ(x − xi ) , | f (xi )| i, f (x )=0, f (x )=0 i ∞ −∞ eiω(t−x) dω, i δ(r) = δ(x − t) = d3 k −ik·r e , (2π )3 ∞ n=0 ϕn∗ (x)ϕn(t) www.elsolucionario.net Curved Orthogonal Coordinates Cylinder Coordinates q1 = ρ, q2 = ϕ, q3 = z; h1 = hρ = 1, h2 = hϕ = ρ, h3 = hz = 1, r = xρ ˆ cos ϕ + yρ ˆ sin ϕ + z zˆ Spherical Polar Coordinates q1 = r, q2 = θ, q3 = ϕ; h1 = hr = 1, h2 = hθ = r, h3 = hϕ = r sin θ, r = xr ˆ sin θ cos ϕ + yr ˆ sin θ sin ϕ + zˆ r cos θ dr = hi dqi qˆ i , A= A·B= A i qˆ i , i i f d3 r = i qˆ A A i Bi, A × B = B1 F · dr = f (q1 , q2 , q3 )h1 h h dq1 dq2 dq3 V L B · da = B1 h h dq2 dq3 + B h1 h dq1 dq3 + qˆ A2 qˆ A3 B2 B3 Fi hi dqi i B h1 h dq1 dq2 , S ∇V = qˆ i i ∂V , hi ∂qi ∇·F = ∂ ∂ ∂ (F1 h h ) + (F2 h1 h ) + (F3 h1 h ) h1 h h ∂q1 ∂q2 ∂q3 ∇2 V = ∂ h 2h ∂ V h1 h h ∂q1 h1 ∂q1 ∇×F = h1 h h + h1 qˆ h qˆ h qˆ ∂ ∂q1 ∂ ∂q2 ∂ ∂q3 h1 F1 h F2 h F3 ∂ h1 h ∂ V ∂q2 h ∂q2 + ∂ h h1 ∂ V ∂q3 h ∂q3 Mathematical Constants e = 2.718281828, π = 3.14159265, ln 10 = 2.302585093, rad = 57.29577951◦ , 1◦ = 0.0174532925 rad, 1 γ = lim + + + · · · + − ln(n + 1) = 0.577215661901532 n→∞ n (Euler-Mascheroni number) 1 1 B1 = − , B2 = , B4 = B8 = − , B6 = , (Bernoulli numbers) 30 42 www.elsolucionario.net Essential Mathematical Methods for Physicists www.elsolucionario.net www.elsolucionario.net Essential Mathematical Methods for Physicists Hans J Weber University of Virginia Charlottesville, VA George B Arfken Miami University Oxford, Ohio Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo www.elsolucionario.net Sponsoring Editor Production Editor Editorial Assistant Marketing Manager Cover Design Printer and Binder Barbara Holland Angela Dooley Karen Frost Marianne Rutter Richard Hannus Quebecor This book is printed on acid-free paper ∞ Copyright c 2003, 2001, 1995, 1985, 1970, 1966 by Harcourt/Academic Press All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777 Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK Harcourt/Academic Press 200 Wheeler Road, Burlington, MA 01803 http://www.harcourt-ap.com International Standard Book Number: 0-12-059877-9 PRINTED IN THE UNITED STATES OF AMERICA 03 04 05 06 07 Q www.elsolucionario.net Contents xix Preface VECTOR ANALYSIS 1.1 Elementary Approach Vectors and Vector Space Summary Scalar or Dot Product 12 Free Motion and Other Orbits 14 1.3 Vector or Cross Product 20 1.4 Triple Scalar Product and Triple Vector Product 29 Triple Scalar Product 29 Triple Vector Product 31 Gradient, ∇ 35 Partial Derivatives 35 Gradient as a Vector Operator 40 A Geometrical Interpretation 42 Divergence, ∇ 44 A Physical Interpretation 45 1.7 Curl, ∇× 47 1.8 Successive Applications of ∇ 53 1.9 Vector Integration 58 Line Integrals 59 1.2 1.5 1.6 v www.elsolucionario.net vi Contents Surface Integrals 62 Volume Integrals 65 Integral Definitions of Gradient, Divergence, and Curl 66 Gauss’s Theorem 68 Green’s Theorem 70 1.11 Stokes’s Theorem 72 1.12 Potential Theory 76 Scalar Potential 76 Gauss’s Law and Poisson’s Equation 82 Gauss’s Law 82 Poisson’s Equation 84 Dirac Delta Function 86 Additional Reading 95 VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS 96 Special Coordinate Systems 97 Rectangular Cartesian Coordinates 97 Integrals in Cartesian Coordinates 98 Circular Cylinder Coordinates 98 1.10 1.13 1.14 2.1 2.2 Integrals in Cylindrical Coordinates 101 Gradient 107 Divergence 108 Curl 110 2.3 Orthogonal Coordinates 113 2.4 Differential Vector Operators 121 Gradient 121 Divergence 122 2.5 2.6 Curl 124 Spherical Polar Coordinates 126 Integrals in Spherical Polar Coordinates 130 Tensor Analysis 136 Rotation of Coordinate Axes 137 Invariance of the Scalar Product under Rotations 141 Covariance of Cross Product 142 Covariance of Gradient 143 www.elsolucionario.net 920 Index Harmonic series, 259 alternating, 270 ratio test, 264 rearrangement, 271–272 regrouping, 259–260 Hausdorff dimension, 875–877 formula See Baker–Hausdorff, 223 Heat flow equation, 471 Heat flow PDE, 708, 760–761 Heaviside expansion theorem, 753 Heaviside step function, 89n Heaviside unit step function, 696, 728, 738, 748 Heisenberg formulation of quantum mechanics, 225 Heisenberg uncertainty principle, 628 Helmholtz, Hermann Ludwig Ferdinand von, 763 Helmholtz’s equation, 628, 759, 762, 768 Helmholtz’s PDE, 763, 776 Helmholtz’s resonator, 633 Hermite, Charles, 642 Hermite differential equation, 452, 462 Hermite equation, 478 Hermite polynomials, 89, 509, 638–650 alternate representations, 645 first few polynomials, 643–644, 646 generating function, 644 orthogonal functions, 486, 492, 508 orthogonality, 646–647 parity, 644–645 raising/lowering operators, 639–642 recurrence relations, 643 Rodridgues representation, 645 simple harmonic oscillator, 638–639 Hermitian conjugates, 246 Hermitian matrix, 207, 209, 214–216 eigenvalues, eigenvectors, 214–218 Hermitian number operator, 641 Hermitian operators, 491–503 degeneracy, 501 orthogonal eigenfunctions, 498–500 properties, 496 quantum mechanics, 492–493 real eigenvalues, 496–497 square wave, 500 High-frequency components, 666, 669 Higher order differential equations, 758 Hilbert, David, 519 Hilbert determinant, 164–165 Hilbert matrix, 221 Hilbert space, 510, 511, 519 Histogram, 803 Holomorphic function, 335n Homogeneous Helmholtz equation, 776 Homogeneous linear equations, 160–161 Homogeneous Lorentz group, 248–254 Homogeneous medium, 838 Homogeneous ODE, 411, 414, 415 Homogeneous PDE, 470 Homomorphism, 234 Hooke’s law constant, 720 Hopf, E., 893 Hopf bifurcation, 893, 899 Hubble’s law, 11 Hulthen ´ wave function, 722 Hydrogen atom, 657–658, 719–720, 790 Hydrogenic momentum wave function, 720, 723 Hyperbolic cosine, 860 Hyperbolic PDE, 470 Hypergeometric functions, 268, 299 Hyperion, 901 Hypocycloid, 836 I Ill-conditional systems, 220–221 Ill-conditioned matrix, 221 Imaginary part, 320 Impulse function, 731 Impulsive force, 733 Incomplete gamma functions, 545 Independence of reference frame, 141 Independent events, 785 Independent random variables, 795 Indicial equation, 443 Inertia matrix, 211–212 Inertia tensor, 147 Inertial frames, 248n Inequality Bessel, 512–513 Cauchy, 347 Chebyshev, 793–794 Schwarz, 513–514 Infinite-dimensional Hilbert space, 519 Infinite products See Product expansion of entire functions, 392–394 Infinite series, 257–317 See also Series Abel’s test, 279–280 addition/subtraction of series, 260–261 algebra of series, 274–276 alternating series, 269–274 asymptotic series, 314–316 Bernoulli functions, 305 Bernoulli numbers, 302–305 binomial theorem, 284–286 convergence See Convergence convergence tests, 262–269 See also Convergence tests www.elsolucionario.net Index divergence, 257 elliptic integrals, 296–302 Euler-Maclaurin integration formula, 306–307 geometric series, 258–259, 261 harmonic series, 259–260 Leibniz criterion, 270 Maclaurin theorem, 283–284 multiplication of series, 275–276 power series, 291–294 Rieman zeta function, 266, 305, 308, 311, 686 series of functions, 276–281 Taylor’s expansion, 281–291 term-by-term differentiation, 280 Weierstrass M test, 278–279 Zeta function, 266, 305, 308, 311, 686 Inhomogeneous Euler ODE, 430–431 Inhomogeneous linear equations, 161–162 Inhomogeneous ODE, 411, 414, 415 Inhomogeneous ODE with constant coefficients, 431–432 Inhomogeneous PDE, 470 Inhomogeneous PDE (Green’s function), 769–776 Initial conditions, 759 Inner product, 485 Integral Bromwich, 746–752 Cartesian coordinates, in, 98 contour, 337–339 convolution, 713–714 cosine, 547 cylindrical coordinates, in, 101–102 definite, 379–384 differentiation of, 461 elliptic, 296–302 error, 548 Euler, 524–526 exponential, 546–548 Fresnel, 398 integration, 491–492 line, 59–62 logarithmic, 547 orthogonality, 584 overlap, 485 Parseval’s relation, by, 716 Riemann, 59 sine, 547 spherical polar coordinates, in, 130 surface, 62–65 volume, 65–66 work, 103 Integral test, 264–266 921 Integral transforms, 689–755 defined, 689 Fourier transform See Fourier transform Laplace transform See Laplace transform linearity, 689–690 momentum representation, 718–721 Integration See also Integral Fourier series, 684–685 integral definitions of gradient/divergence/curl, 66–67 Laplace transform, 738–739 power series, 292 vector, 58–68 Integration interval, [a,b], 491–492 Intermittency, 902 Internal force, 20 Intersection, 784 Introduction to Quantum Mechanics (Griffiths), 630 Invariant, 148 Invariant subgroup, 231 Inverse cosine transform, 696 Inverse Fourier transform, 694–698, 709 Inverse Laplace transform, 746–754 Inverse operator, uniqueness of, 690 Inverse polynomial, 381–382 Inverse transform, 726 Inversion, 361–363 calculus of residues, 748 mapping, 361–363 matrix, 184–189 numerical, 729 PDE, 708–709 power series, 293–294 square pulse, 696 Inversion of power series, 293–294 Irreducible, 235 group representations, 234–236 Irregular/regular singularities, 448–450 Irregular singular point, 440 Irregular solution, 460 Irrotational, 50, 51 Isolated singular point, 372 Isomorphism, 234 Isoperimetric, 860 Isospin, 229 Isotropic, 146 J Jacobi, Carl Gustav Jacob, 120 Jacobi-Anger expansion, 609 Jacobi identity, 189 Jacobi identity for vector products, 34n Jacobi technique, 223 Jacobian, 119, 742, 800 www.elsolucionario.net 922 Index Jacobians for polar coordinates, 119–120 Jensen’s theorem, 401, 402 Joining conditions, 489 Jordan’s lemma, 383 Julia set, 877 K Kaon decay, 253 Kepler’s first law, 105 Kepler’s law in cylindrical coordinates, 104 Kepler’s orbit equation in polar coordinates, 106 Kirchhoff diffraction theory, 345 Kirchhoff’s laws, 159, 331, 421, 735 Klein-Gordon equation, 757 Koch curve, 876 Kronecker, Leopold, 141 Kronecker delta, 140, 141, 146, 180 Kronecker product, 182 Kronig-Kramers optical dispersion relations, 751n L l’Hopital’s rule, 288, 546, 612 ˆ Ladder operator, 245 Ladder operator approach, 244–247 Lagrange, Joseph Louis comte de, 40 Lagrange multiplier, 212 Lagrange multiplier method, 39–40 Lagrange undetermined multiplier, 849 Lagrangian, 841 Lagrangian equations of motion, 841–843, 853 Lagrangian multipliers, 848–852 Laguerre, Edmond Nicolas, 655 Laguerre differential equation, 462 Laguerre functions, 657 Laguerre ODE, 452 Laguerre polynomials, 508, 650–661 associated, 486, 492, 508, 655–658 generating function, 652 hydrogen atom, 657–658 Laguerre’s ODE, 650–652 lowest polynomials, 653 normalization, 509, 654 orthogonal functions, 486, 492, 508 orthogonality, 654–655 recursion relation, 653–654 Rodrigues’s formula, 653 special integrals, 655 Lane-Emden equation of astrophysics, 469 Langevin theory of paramagnetism, 294 Laplace, Pierre Simon, 54 Laplace convolution, 742–746 Laplace equation, 84, 571, 572, 680, 756, 759, 846 Laplace equation of electrostatics, 54 Laplace PDE, 846 Laplace series, 588 Laplace transform, 693, 724–752 Bessel’s function, 421 Bromwich integral, 746–752 convolution theorem, 742–746 defined, 724 derivative of a transform, 737–738 derivatives, of, 730–734 Dirac delta function, 732 Euler integral, 693 integration of transforms, 738 inverse, 746–754 inverse transform, 726 numerical inversion, 729 operational theorems, 752 partial fraction expansion, 726–727 RLC analog, 735–736 substitution, 734 translation, 736 Laplacian, 54–55 Laplacian development by minors, 164 Laurent, Pierre-Alphonse, 357 Laurent expansion, 350–359 Laurent expansion by integrals, 356–357 Laurent expansion by series, 357 Laurent series, 354–358, 677 Law area law for planetary motion, 104–107 Bayes’s decomposition, 796 Biot-Savart, 35 conservation, 229 cosines of, 19, 558 decay, 412 Faraday’s, 74 Faraday’s induction, 56, 74 Gauss’s, 82–84 Hubble’s, 11 Kepler’s, 104, 105 Kirchhoff’s, 159 Malthus’s, 879 Oersted’s, 57, 74 Snell’s law of refraction, 836 Stefan-Boltzmann, 539 triangle law of vector addition, Law of cosines, 19, 558 Law of sines, 28 Laws of large numbers binomial distribution, 797–799, 802–806 continuous Gauss distribution, 810 Gauss distribution, 809 Poisson distribution, 806 www.elsolucionario.net Index Legendre, Adrien Marie, 558 Legendre differential equation, 461 self-adjoint form, 565 singularities, 441, 452 Legendre duplication formula, 528, 625 Legendre equation, 451, 487, 565 self-adjoint form, 565 Legendre ODE, 565, 568 Legendre polynomials, 276, 534, 552–588 associated, 486, 492, 581–586 differential equations, 564–565 electric multipoles, 558–559 electrostatic potential of ring of charge, 573–574 generating function, 553–555 Gram-Schmidt orthogonalization, 506–507 Legendre series, 569–570 Legendre’s ODE, 565, 568 linear electric multipoles, 558–559 lowest, 557, 579 orthogonal functions, 486, 492, 508 orthogonality, 568–579 parity, 556 physical basis (electrostatics), 552–553 polarization of dielectric, 577 power series, 556 recurrence relations, 563–564 Rodrigues’s formula, 579 shifted, 486, 492, 508, 509 sphere in uniform field, 570–573 spherical harmonics, 584–586 table of functions, 554 upper/lower bounds for Pn , 566 uses, 552 vector expansion, 559–560 Legendre series, 269, 569–570 Leibniz, Gottfried Wilhelm von, 172, 318 Leibniz criterion, 270 Leibniz differentiation formula, 582 Leibniz formula for derivative of an interval, 461 Lemma Jordan’s 383 Riemann’s 691 Levi-Civita symbol, 153–157 l’Hopital’s rule, 288, 546, 612 ˆ Lie, Sophus, 233 Lie algebra, 237 Lie groups, 230 Limit cycle, 892 Limit tests, 267 Line integral, 59–62 Line integral for work, 59–61 923 Linear electric multipoles, 558–559 Linear electric octopole, 561 Linear electric quadrupole, 559, 561 Linear equations, 159–162, 167–168 Linear first-order ODEs, 414–418 Linear ODE, 410 Linear space, Linearly independent solution, 416 Liouville, Joseph, 501 Liouville’s theorem, 348 Local bifurcations, 898 Logarithmic integral, 547 Logistic map, 869–874 Lorentz, Hendrik Antoon, 254 Lorentz covariance, 248 Lorentz-Fitzgerald contraction, 155, 254 Lorentz force, 22 Lorentz frame, 148 Lorentz gauge, 57 Lorentz group, 254 Lorentz invariant energy squared, 252 Lorentz invariant infinitesimal version, 251 Lorentz invariant momentum transfer squared, 253 Lorentz line shape, 745n Lorentz scalar, 249 Lorentz transformations, 147, 156, 248–251 Lorenz coupled NDEs, 902 Lowering operator, 245, 639–642 Lowest associated Legendre polynomials, 583–584 Lowest Legendre polynomials, 557, 579 Lowest spherical harmonics, 585–586 Lyapunov exponent, 874–875, 900 M Maclaurin, Colin, 286 Maclaurin expansion, 537 Maclaurin expansion of the exponential, 227 Maclaurin integral test, 264–266 Maclaurin series, 46, 723 Maclaurin theorem, 283–284 Madelung constant, 276 Magnetic field, 48, 56–58, 71, 73–76, 81, 102, 103–104, 111–112, 147 Magnetic flux, 102–104 Magnetic induction of long wire, 111–112 Magnitude, 322 of scalar, of vector, Malthus’s law, 879 Mandelbrot set, 877 Maple, 171, 450, 465, 537, 548 www.elsolucionario.net 924 Index Mapping, 360–368 branch points/multivalent functions, 363–367 conformal, 368–370 inversion, 361–363 rotation, 361 translation, 360 Marginal probability distribution, 796 Matching conditions, 489 Mathcad, 171, 537, 548 Mathematica, 171, 450, 465, 537, 548 Mathematical manipulation computer software, 171 Mathematical Methods for Physicists (Arfken, Weber), 50, 96 Mathematicians See Biographical data Mathematics of Computation, 622 Matlab, 171 Matrices, 174–228 addition/subtraction, 179–180 anti-Hermitian, 216 defined, 174 determinant of matrix product, 181–182 diagonal, 182–183 diagonalization, 211–228 See also Diagonalization of matrices direct product, 182 equality, 175 Euler angle rotation, 200–201 Gauss-Jordan matrix inversion, 186–188 Hermitian, 207, 209, 214–216 Hilbert, 221 ill-conditioned, 221 inertia, 211–214 inversion, 184–189 ladder operators, 244–247 minor, 164–165 multiplication, 175–180 null, 180 orthogonal, 193–204 See also Orthogonal matrices Pauli, 208–209 product theorem, 180–182 rank, 175 representation, 234–236 similarity transformation, 202–204 singular, 224 symmetry, 146–147, 202 trace, 184 transpose, 202, 204 transposition, 178 unit, 180 unitary, 207, 209 vector transformation, 136–144, 176, 184, 188, 193–204, 207 Matrix addition, 179–180 Matrix inversion, 184–189 Matrix multiplication, 175–180 Maximum surface tension, 852 Maxwell-Boltzmann (MB) statistics, 788 Maxwell’s equations, 56–57 Lorentz covariance of, 248 partial differential equations, 48, 55–57, 73–74, 82–85 McMahon’s expansion, 908 Mean value, 791 Mean value theorem, 282, 347n M´ecanique Analytique (Lagrange), 40 Mellin transform, 729 Meromorphic function, 372, 390 Method of least squares, 792 Method of steepest descents, 400–409 Methods of Mathematical Physics (Courant/Hilbert), 758 Metric, 115 Minkowski space, 121, 144, 249 Minkowski space-time, 251–254 Mittag-Leffler theorem, 390–392 Mixed tensors, 144 Modified spherical Bessel functions, 778 Modulus, 322 Moment generating function, 794 Moment of inertia, 63–64 Moment of inertia ellipsoid, 213 Moment of inertia matrix, 211–212 Momentum representation, 718–721 Momentum wave equation, 721 Monopole, 559 Morera’s theorem, 346–347 Mott cross section, 701 Movable singularity, 881–882 Moving particle, 842–843 Multiplet, 235 Multiplication complex number, 321 complex variable, by, 324–325 matrix, 175–180 scalar, 9, 10 series, 275–276 vector See Vector multiplication Multipole expansion, 560 Multivalent functions, 363–367 Multivalued function, 326, 374 Mutually exclusive, 782, 783 N N-fold degenerate, 501 Navier-Stokes equation, 55 NDEs See Nonlinear differential equations (NDEs) www.elsolucionario.net Index Negative of a vector, Neumann, Karl, 613 Neumann boundary conditions, 634, 758, 759 Neumann functions, 611–617 See also Spherical Neumann functions Y0 (x), 460, 612 Y1 (x), 612 Y2 (x), 612 Yv (x), 612, 616 Neumann problem, 767 Neumann series solution (quantum mechanical scattering), 771–774 Neutrino energy density (Fermi distribution), 539 Neutron transport theory, 295 Newton’s equation for a single particle, 252 Newton’s equations of motion, 20, 117, 842 Newton’s formula for finding a root, 287 Newton’s law of universal gravitation, 84 Newton’s method, 905 Newton’s second law, 731, 733 Node See sink, attractor, 870 Noether’s theorem, 229 Nonlinear differential equations (NDEs), 878–903 autonomous differential equations, 882 Bernoulli equations, 879–880 bifurcations in dynamic systems, 898–899 chaos in dynamic systems, 900–902 dissipation in dynamic systems, 896–898 fixed singularities, 881–882 local/global behavior in higher dimensions, 885–896 Poincare´ section, 878 Riccati equations, 880–881 routes to chaos in dynamic systems, 901–902 special solution, 882 Verhulst’s NDE, 882, 883 Nonperiodic function, 694 Nonuniform convergence, 277–278 Normal distribution, 807–812 Normal modes of vibration, 218–220 Normalization, 600 Bessel functions, 600 Gram-Schmidt orthogonalization, 503–506 Hermite polynomials, 509 hydrogen momentum wave function, 723 Laguerre polynomials, 509, 654 momentum representation, 719 orthogonal polynomials, 508 Normalized hydrogen wave function, 658 Normalized simple harmonic oscillator wave function, 649 Nuclear stripping reactions, 636 925 Null matrix, 179 Null vector, Number operator, 641 Numerical Recipes, 622 Numerical solutions (ODE), 467–470 O O(n), 232, 233 Odd permutation, 155n Odd symmetry, 445 ODE, 410 See also Differential equations ODE with constant coefficients, 428–429 Oersted’s law, 57, 74, 103, 104, 111 Ohm’s law, 72, 73, 159 Olbers’s paradox, 268 One-dimensional, time-independent Schrodinger equation, 723 See also ă Schrodinger (wave) equation ă One-sided Laplace transform, 724n Operator adjoint, 484 curl, 47–51, 110–112, 124–125, 129 divergence, 44–47, 108–109, 122–123, 129 gradient, 41–43, 107–108, 121–122, 129 Hermitian See Hermitian operators lowering, 639–642 number, 641 projection, 505 raising, 639–642 self-adjoint, 484 Optical dispension, 750 Optical path near event horizon of black hole, 831–832 Orbital angular momentum, 243–248 ladder operator approach, 244–247 rotation of functions, 239–240 Orbital angular momentum equation, 586 Orbital angular momentum in cylindrical coordinates, 118–119 Order (determinants), 163 Order of a Lie group, 237 Order of matrix, 175 Ordinary differential equation (ODE), 410 See also Differential equations Ordinary point, 439, 440 Orthogonal azimuthal functions, 587 Orthogonal coordinates, 113–121 Orthogonal eigenfunctions, 498–500 Orthogonal functions, 482–522 eigenvalue/eigenvector See Eigenvalue/eigenvector Gram-Schmidt orthogonalization, 503–510 Hermitian operators See Hermitian operators Legendre polynomials, 506–507 www.elsolucionario.net 926 Index Orthogonal functions (cont.) orthogonal polynomials, 507–508 self-adjoint ODEs, 483–496 Orthogonal matrices, 193–204 direction cosines, 194–195 Euler angles, 200–201 similarity transformation, 204 symmetry, 202 tensors, and, 204 two-dimensional case, 198–200 vectors, and, 195–198 Orthogonal polynomials, 507–508 Orthogonality, 498 Bessel functions, 599–600 Fourier series, 499, 664 Hermite polynomials, 646–647 Laguerre polynomials, 654–655 Legendre polynomials, 568–579 vectors, 483 Orthogonality integral, 584 Orthonormal, 503 Oscillatory series, 261 Overlap integral, 485 Overshoot (Gibbs phenomenon), 666 P Parabolic PDE, 470 Parachutist, 412–413, 896 Parallelogram addition, Parallelogram of forces, 6–7 Parallelepiped, 30–31 Parity associated Legendre polynomials, 584 even, 446 Fourier sine/cosine transform, 703 Hermite polynomials, 644–645 Legendre polynomials, 556 spherical harmonics, 586 Parseval’s identity, 517, 521, 686–687 Parseval’s relation, 715–716 Partial derivatives, 35, 470 of a plane, 36–37 Partial differential equations (PDEs), 470, 756–781 alternate solutions, 763–765 boundary conditions, 758–760 diffusion PDE, 760–761 examples of PDEs in physics, 756–757 first-order PDEs, 758 heat flow PDE, 708, 760–761 Helmholtz’s PDE, 763, 776 inhomogeneous PDE, 769–776 See also Green’s function inversion of PDE, 708–709 Poisson’s PDE, 709 solving second-order PDEs, 757–758 terminology, 470 Partial fraction expansion, 726–727 Partial sum, 257 Particle in a box, 850 Particle spin, 147n Particular solution, 460 Particular solution of homogeneous ODE, 416 Pascal, Blaise, 825 Passive transformation, 202 Path-dependent work, 59 Pauli, Wolfgang, 209 Pauli exclusion principle, 859 Pauli spin matrices, 190, 208–209 Pauli theory of the electron, 58 PDEs See Partial differential equations (PDEs) Pendulum damped, 892 periodically driven, 902 simple, 296–297, 853–854 Period doubling, 901–902 Period of cycle, 871 Periodically driven pendulum, 902 Permutations, 155n, 787–788 Phase, 323 Phase shift, 630 Phase space, 91, 868 Phase transition, 875 Physicists See Biographical data Piecewise regular, 664 π mesons, 147 Pi meson (pion), 147 Pion photoproduction threshold, 253 Pitchfork bifurcation, 871, 899 Planck’s black-body radiation law, 311 Planck’s theory of quantized oscillators, 289 Plane circular membrane, vibrations, 600–604 Pochhammer symbol, 538 Poincare, ´ Jules Henri, 867, 869 Poincare-Bendixson theorem, 896 ´ Poincare´ group, 248 Poincare´ section, 878, 879, 900 Poisson, Simeon ´ Denis, 85 Poisson distribution, 804–806, 810 Poisson’s equation, 84–85, 756 Poisson’s PDE, 709 Pole, 373 Pole expansion of cotangent, 391–392 Pole expansion of meromorphic functions, 390–391 Polygamma function, 536 Polynomials associated Legendre, 486, 492, 581–586 associated Laguerre, 486, 492, 508, 655–658 www.elsolucionario.net Index Hermite See Hermite polynomials Laguerre See Laguerre polynomials Legendre See Legendre polynomials orthogonal, 507–508 shifted Legendre, 486, 492, 508, 509 Potential, 41 centrifugal, 79–80 of conservative force, 76–80 Coulomb, 45 gravitational, 61–62 scalar, 76–78 vector, 55 Potential theory, 76–81 Power series, 291–294 continuity, 292 convergence, 291 differentiation/integration, 292 inversion of, 293–294 Legendre polynomials, 556 matrix, of, 221–222 uniqueness theorem, 292–293 Power series expansions differential equations (Frobenius’s method), 441–454 erf z, 548 exponential integral, 547–548 Power spectrum, 670 Predictor-corrector methods, 467–468 Prime number theorem, 273 Principal axes, 212–216 Principal value, 326 Probability, 782–825 basic principles, 783 binomial distribution, 802–804, 810 central moments, 794 Chebyshev inequality, 793–794 coin tossing, 783 conditional, 785 correlation, 795 covariance, 795 Gauss distribution, 807–812 marginal, 796 method of least squares, 792 normal distribution, 807–812 permutations and combinations, 787–788 Poisson distribution, 804–806, 810 random variables, 789 repeated draws of cards, 796–799 repeated tosses of dice, 802 simple properties, 782–786 standard deviation, 792 statistics, 812–825 See also Statistics sum, product, ratio of random variables, 800–801 variance, 793 927 Probability amplitudes, 521 Product convergence theorem, 275 Product expansion of entire functions, 392–394 Product theorem for determinants, 180–182 Projection operator, 505, 522 Proton charge form factor, 701–703 Pseudotensor, 153 Pseudovector, 142 Pythagorean theorem area law, 105 Cartesian coordinates, 97 scalar product, 19 vector, Q Quadrupole, 560 Quadrupole moment tensor, 587 Quadrupole tensor, 147 Quantization of angular momentum, 452 Quantum chromodynamics, 703 Quantum mechanical angular momentum operators, 587 Quantum mechanical particle in a sphere, 628 Quantum mechanical scattering, 387–390, 771–776 Quantum mechanical simple harmonic oscillator See Simple harmonic oscillator Quantum theory of atomic collisions, 397 Quark counting, 703 Quasiperiodic motion, 893 Quasiperiodic route to chaos, 902 Quaternions, 139n Quotient rule, 151–153 R Radial ODE, 487 Radial Schrodinger wave equation, 463 ă Radial wave functions, 486, 630 Radioactive decay, 412 Raising operator, 245, 639–642 Raleigh expansion, 636 Random event, 782 Rank (matrix), 175 Rapidity, 251 Ratio test, 262–263 Rational functions, 390–391 Rayleigh, John William Strutt, Lord, 861 Rayleigh equation, 580, 778 Rayleigh formulas, 629 Rayleigh plane wave expansion, 574 Rayleigh-Ritz variational technique, 861–865 Rayleigh’s theorem, 715n Real eigenvalues, 496–497 Real part, 320 www.elsolucionario.net 928 Index Real zeros of a function, 905–909 Reciprocal cosine, 380–381 Rectangular Cartesian coordinates, 97–98 Recurrence relation Hermite polynomials, 643 Laguerre polynomials, 653–654 Legendre polynomials, 563–564 Neumann function, 613 spherical Bessel function, 629 two-term, 443 Recursion formula, 564 Reduce, 171, 450, 465, 537, 548 Regression coefficient, 815 Regular function, 335n Regular/irregular singularities, 448–450 Regular singular point, 440 Regular solution, 460 Relativistic particle, Lagrangian, 844 Relaxation methods, 223 Repeated draws of cards, 796–799 Repelling spiral saddle point, 894–896 Repellor, 870, 883, 884 Residue theorem, 378–379, 394 Resistance-inductance circuit, 418 Resonance condition, 745 Resonant cavity, 604–608 Riccati equations, 880 Riccati NDE, 880 Richardson’s extrapolation, 467 Riemann, Bernhard Georg Friedrich, 335 Riemann integral, 59, 498n Riemann lemma, 691 Riemann surface, 366, 367 Riemann theorem, 272 Riemann zeta function Bernoulli numbers, 305 comparison test, 262–263 convergence, 308–309 Maclaurin integral test, 266–267 polygamma function, 536 table of values, 313 Riemannian, 115 Riemann’s lemma, 691 Riemann’s theorem, 272 Right-hand rule for positive normal, 63 Ring, 180 RL circuit, 418 RLC analog, 735–736 RLC circuit, 432, 735 Rocket shortest distance between two rockets in free flight, 24–25 shortest distance of observer from, 840–841 shortest distance of rocket from observer, 17–20 Rodrigues’s formula/ representation, 579 associated Laguerre polynomial, 656 Hermite polynomials, 645 Laguerre polynomials, 653 Root test, 262 Rosetta curves, 107 Rossler coupled ODEs, 902 ă Rotation, 361 Rotation ellipsoid, 131–133 Rotation groups SO(2) and SO(3), 238–239 Rotation of functions/orbital angular momentum, 239–240 Rotation-soap film problem, 832–834 Rouche’s ´ theorem, 393, 394 Row vector, 178 Rule addition, 784 BAC-CAB, 32 chain See Chain rule Cramer’s, 162 l’Hopital’s, 288 ˆ quotient, 151–152 right-hand rule for positive normal, 63 Runge-Kutta method, 466467 Runge-Kutta-Nystrom ă method, 468 S S-wave particle in an infinite spherical square well, 863–864 Saddle point, 336, 401, 888–889, 898 Saddle point axis, 404 Saddle point in one dimension, 884 Saddle point method, 402–407 Sample, 783 SAT tests, 785–786 Sawtooth wave, 665–666 Scalar Kirchhoff diffraction theory, 779 Scalar multiplication, 9, 10 Scalar potential, 76–78 Scalar product of vectors, 12–20 Scalar or inner product of functions, 483, 485, 498, 516 Scalar quantities, Scale factors, 116 Scattering theory of relativistic particles, 91 Schlaefli integral, 618 Schrodinger equation potential, 634 ă Schrodinger (wave) equation, 206 ă central potential, 865 constrained minimum, 856–857 deuteron, 488 eigenvalue problems, 223 hydrogen atom problem, 478 matrix representation, 234 momentum representation, 718 www.elsolucionario.net Index one-dimensional, time-independent equation, 723 PDEs, 757 quantum mechanical scattering, 771 single particle system, 485 Schwarz inequality, 337, 513–514 Schwarz reflection principle, 351–352, 538 Schwarz’s theorem, 358 Second-order ODEs, 424–439 missing variable x, 426 missing variable y, 425–426 Second-order PDEs, 470, 757–758 Second-rank tensor, 144–145 Second solution (ODE), 454–464 Secular equation, 214 Self-adjoint, 207, 484 Self-adjoint ODEs, 483–496 Self-similar sets, 875 Separation of variables for ODEs and PDEs, 411–412, 470–480 Cartesian coordinates, 471–473 circular cylindrical coordinates, 474–476 first-order ODE, 411–412 spherical polar coordinates, 476–478 Series addition/subtraction, 260–261 algebra of, 274–276 alternating, 269–274 asymptotic, 314–316 Bessel, 600 Fourier See Fourier series geometric, 258–259 harmonic.See Harmonic series infinite See Infinite series Laurent, 354–358 Legendre, 269, 569–570 oscillatory, 261 power, 291–294 Stirling’s, 541–542 Taylor See Taylor expansion Series solutions (Frobenius’s method), 441–454 Series summation, 537 Servomechanism, 745 Shifted Legendre polynomials, 486, 492, 508, 509, 729 SHO equation, 451 Shortest distance of observer from rocket, 840–841 Similarity transformation, 204, 231–233 Simple harmonic oscillator driven harmonic oscillator, 707–708 Hermite polynomials, 638–639 momentum representation, 720–721 929 normalized wave function, 649 orthogonal functions, 486, 492 3-D, 661 Simple harmonic oscillator (SHO) equation, 451 Simple pendulum, 296–297, 853–854 Simple pole, 373 Simple unitary groups, 233 Sine integral, 547 product representation, 392 Sine transform, 700 Single harmonic oscillator, classical harmonic oscillator, 731–732 Single well, 899 Singular, 224 Singular points, 439–441 Singular solution, 419 Singularities, 372–377 Bessel, 441 branch points, 374, 377 essential, 373, 440 fixed, 881–882 movable, 881–882 poles, 373–374 regular/irregular, 448–450 Singularity on contour of integration, 386–387 Sink, 870, 883, 884 Skewsymmetric matrix, 202 SL(2), 236 Sliding off a log, 854–856 Snell’s law of refraction, 836 SO(2), 231, 238–239 SO(3), 233, 238–239 SO(n), 232, 233 Soap film, 832–834 Solenoidal, 51 Source, 414 Special linear group SL(2), 236 Special relativity, 249 Special unitary group SU(2), 240–242 Spherical Bessel functions, 624–637 definitions, 624–627 limiting values, 627–628 numerical computation, 630 recurrence relations, 629 spherical Hankel functions, 624, 627 spherical Neumann functions, 624, 626 Spherical Green’s function, 778 Spherical Hankel functions, 624, 627 Spherical harmonic closure relation, 587 Spherical harmonics, 584–586 Spherical Neumann functions, 624, 626 Spherical polar coordinates, 126–136, 760 Spherical symmetry, 225 Spherically symmetric heat flow, 766–767 www.elsolucionario.net 930 Index Spin particle, 242 Spin states, 247 Spin wave functions, 240, 241 Spin particles, 147 Spin particles, 147 Spin zero particles, 147 Spinor wave functions, 209 Spinors, 147 Spiral fixed point, 890–891 Spiral repellor, 889, 898 Spiral saddle point, 894 Spiral sink, 889, 898 Spontaneous or movable singularity, 881 Square pulse, 691–692, 696 Square wave, 500, 511 Square wave-high frequencies, 668–669 Stable sink, 886–888 Standard deviation, 792 Standing spherical waves, 627 Stark effect, 453, 661 Statistical hypothesis, 812 Statistical mechanics, 787–788 Statistics, 812–825 See also Probability Chi squared distribution, 817–821 confidence interval, 823–824 error propagation, 812–815 fitting curves to data, 815–817 student t distribution, 821–823 Steady-state current, 433 Steepest descent, method of, 400–409 Stefan-Boltzmann law, 539 Step function, 727–728 Stieltjes integral, 89n Stirling’s asymptotic formula, 268, 274, 810 Stirling’s expansion of the factorial function, 407 Stirling’s formula, 316, 408, 542, 543, 806, 811, 859 Stirling’s series, 541–542, 543, 544 Stokes, Sir George Gabriel, 75 Stokes’s theorem, 72–73 Bessel functions, 617, 623 Cauchy’s integral theorem, 339–341 curl (cylindrical coordinates), 110, 124 magnetic flux, 103–104 Straight line, 830–831 Strange attractor, 877 Structurally stable fixed points, 882 Structurally unstable fixed points, 882 Structure constants of a Lie group, 237 Structure of the Nucleus (Preston), 331 Student t distribution, 821–823 Sturm, Jacques Charles, 501 Sturm-Liouville equation, 861 Sturm-Liouville theory, 485, 569 See also Orthogonal functions SU(2), 233, 240–242 SU(n), 233 Subgroup, 231 Substitution (Laplace transform), 734 Subtraction complex number, 321 matrix, 179–180 series, 260–261 tensors, 145 vector, Subtraction of sets, 784 Sufficiency conditions, 829n Summation convention (tensor analysis), 145–146 Summing series, 537 Superposition principle, 411 Euler’s ODE, 427–428, 429 homogeneous PDEs, 470 linear electric multipoles, 558 ODEs with constant coefficients, 428–429 separation of variables, 473 vibrating membranes, 603 Surface integrals, 62–65 Surface of Hemisphere, 130 Symmetric matrix, 202 Symmetric tensor, 147 Symmetry azimuthal, 571, 586 ODEs, 445–446 periodic functions, 671 tensors, 146–147 T Table of the Gamma Function for Complex Arguments, 543 Tables of Functions (Jahnke/Emde), 476 Taking the complex conjugate, 321 Tangent bifurcation, 903 Taylor, Brooke, 282–283 Taylor expansion, 136–157, 350–351, 672 Taylor series solution, 464–465 Tensor analysis, 136–157 addition/subtraction of tensors, 145 contraction, 149 contravariant tensor, 144–145 covariance of cross product, 142 covariance of gradient, 143 covariant tensor, 144–145 direct product, 149–151 dual tensors, 153–157 Levi-Civita symbol, 153–156 www.elsolucionario.net Index pseudotensor, 153 quotient rule, 151–153 rotation of coordinate axes, 137–142 spinors, 147 summation convention, 145–146 symmetry-antisymmetry, 146–147 tensors of rank two, 144–145 Tensor of rank zero, 136 Tensor of rank one, 136 Tensor of rank two, 144–145 Tensor of rank n, 136 Tensor transformation law, 115n Term-by-term differentiation, 280 Theorem Abel’s, 678 addition, 594 binomial, 284–286 Cauchy’s integral, 337–343 center manifold, 898 Fuchs’s, 450 Gauss’s, 68 Green’s, 70 Jensen’s, 401, 402 Liouville’s, 348 Maclaurin, 283–284 mean value, 282 Morera’s, 346–347 Poincare-Bendixson, 896 ´ prime number, 273 product, 180–182 residue, 378–379 Riemann’s, 272 Rouche’s, ´ 393 Schwarz’s, 358 Stokes’s, 72–73 uniqueness, 292–293 Theory of anomalous dispersion, 751n Theory of free radial oscillations, 907 Three-dimensional harmonic oscillator, 661 Time-dependent scalar potential equation, 757 Time-dependent Schrodinger equation, 206 ă See also Schrodinger (wave) equation ¨ Time-independent diffusion equations, 756 Time-independent Schrodinger equation, 234, ¨ 723, 771 Total charge inside a sphere, 90 Total variation of a function, 37 Trace, 184 Trace formula, 222 Trajectory See also orbit, 14, 17–18, 105–106, 868, 878, 882–896 Transfer function, 745 Transformation theory, 136 Transformations, 202–204 931 Translation, 360, 736 Translation invariance, 4n Transpose, 178 Transposed matrix, 202, 204 Transposition, 178 Transversality condition, 839–841 Traveling spherical waves, 627 Treatise on Algebra (Maclaurin), 162 Triangle inequalities, 324 Triangle law of vector addition, Trigonometric Fourier series, 677 Triple scalar product, 29–31 Triple vector product, 31–33 Two-index Levi-Civita symbol, 157 Two-sided Laplace transform, 724n U 238 U, 807 U(1), 233 U(n), 233 Undershoot (Gibbs phenomenon), 666 Uniform convergence, 276–277 Union of sets, 784 Uniqueness theorem, 292–293 Unit matrix, 180 Unit step function, 395, 687, 738–739 Unit vector, Unitary groups, 233 Unitary matrix, 207–209 Universal Feigenbaum numbers, 873, 901 Universal number, 875 V Van der Pool nonautonomous system, 893 Variance, 793 Variation of the constant, 416 Variation with constraints, 850–852 Variational principles See Calculus of variations Vector See also Vector analysis contravariant, 143 covariant, 143 defined, differentiation, 17 direct product, 150–151 geometric representation, negative of, null, orthogonal matrices, 195–198 physics, and, row, 178 unit, uses, Vector addition, 2–3 www.elsolucionario.net 932 Index Vector analysis, 1–158 See also Vector BAC-CAB rule, 33 cross product, 20–24 curl, 47–51 curved coordinates See Curved coordinates Dirac delta function, 86 divergence, 44–47 elastic forces, 8–9 electromagnetic wave equations, 56–57 elementary approach, 1–12 free particle motion, 14–17 Gauss’s law, 82–84 Gauss’s theorem, 68–69 gradient, 35–44 See also Gradient Green’s theorem, 70 medians of triangle meet in center, 25–26 Minkowski space-time, 251–253 parallelogram of forces, 6–7 Poisson’s equation, 84–85 potential theory, 76–81 scalar potential, 76–78 scalar product, 12–20 shortest distance between two rockets in free flight, 24–25 shortest distance of rocket from observer, 17–20 Stokes’s theorem, 72–73 successive applications of the gradient operator, 53–58 tensor analysis See Tensor analysis triple scalar product, 29–31 triple vector product, 31–33 vector addition, 2–3 vector integration, 58–68 vector potential of constant B field, 48 vector subtraction, Vector equality, Vector field, Vector integration, 58–68 Vector multiplication cross product, 20–24 direct product, 149–151 scalar product, 12–20 Vector potential, 55 Vector product, 20–24 Vector space, Vector subtraction, Velocity of electromagnetic waves, 749–752 Verhulst, P F., 869 Verhulst’s NDE, 882, 883, 900 Vibrating membranes, 600–604 Vibrating string, 863 Volterra integral equations, 743 Volume ellipsoid, 131 rotated Gaussian, 66 Volume integrals, 65–66 von Staudt-Clausen theorem, 304 Vorticity, 50n W Wave equation, 709–711 Weierstrass, Karl Theodor Wilhelm, 528 Weierstrass infinite product definition, 526–528 Weierstrass M test, 278–279 Weight function, 485–487 Wigner, Eugen Paul, 233 WKB expansion, 314 Work integral, 103 Wronski, Jozef Maria, 436 ´ Wronskian Bessel functions, 613–614 linear first-order ODE, 417 second-order ODEs, 435, 436 second solution, ODE, 454–456 uses, 614 Y Yukawa potential, 778 Z Zeeman effect, 216n, 235 Zero-point energy, 639 Zeros, of functions, 392, 905–908 of Bessel functions, 598–599 Zeta function See Riemann zeta function www.elsolucionario.net Series and Products f (x) = ∞ f (n) (a) n=0 (x − a)n , n! f (z) = f (0) + ∞ bn n=0 1 + z − an an ∞ ∞ f f z z/a n f , f (z) = f (0)e z f (0) 1− e + (z) = (0) + f f z − a a a n n n n=0 n=1 = 1−x ln(1 + x) = ∞ n=0 ∞ (−1)n−1 n x n ∞ (−1)n x 2n+1 , (2n + 1)! n=0 ex ∞ xn α n x x , e = , n n! n=0 n=0 n=1 sin x = ∞ x n, (1 + x)α = cos x = ∞ n=0 (−1)n x 2n , (2n)! ∞ x x 2n x B 2n = 1− + −1 n=1 (2n)! x cot x = ∞ (−1)n B 2n n=0 ζ (s) = ∞ , ns n=1 π = sin2 π x (2x)2n , (2n)! ζ (2n) = (−1)n−1 ∞ , n=−∞ (x − n) ∞ (z) = e =e Jν (x) = −y ∞ n=0 1 + x−n x+n 1− x2 n2 ∞ x −x/n 1+ = xeγ x e , (x) n n=1 1 − n z+ n n=1 iz ∞ n=1 ∞ ∞ + x n=1 (2π )2n B 2n 2(2n)! sin π x = π x e−t t z−1 dt, (z + 1) = −γ + cot π x = (cos x + i sin x), x (−1) n!(ν + n)! (1 − 2xt + t )−1/2 = ln z = ln |z| + i(arg z + 2π n) ν+2n n , e(x/2)(t−1/t) = ∞ Jn(x)t n n=−∞ ∞ Pl (x)tl , Pl (x) = d 2l l! dx Pl (x)Pl (x)dx = 2δll 2l + l=0 −1 l (x − 1)l , www.elsolucionario.net eik·r = 4π e −t +2tx bn = l ∗ Ylm (θk , ϕk )Ylm(θ, ϕ) il jl (kr) m=−l l=0 ∞ tn = Hn(x) , n! n=0 Ln(x) = f (x) = ∞ Hn(x) = (−1)ne x ∞ e x dn n −x (x e ), n! dx n dn −x e dx n e−x Lm(x)Ln(x)dx = δmn ∞ a0 (a n cos nx + bn sin nx), a n = + π n=1 π 2π f (x) cos nx dx, 2π f (x) sin nx dx Integrals ∞ F(ω) = √ 2π ∞ −∞ ∞ −∞ F(ω)G ∗ (ω)dω = g(y) f (x − y)dy = g(z)e dz ∼ ∇2 V = − = A B E Z H I K M α β γ δ ζ η θ ι κ λ µ ∞ F(ω)e−iωt dω −∞ f (t)g ∗ (t)dt, F(ω)G(ω)e−iωx dω 2π g(z0 )es f (z0 )+iα π , f (z0 ) = 0, α = − arg f (z0 ) 1/2 |s f (z0 )| 2 ρ , ε0 V (r) = 4π ε0 e−mr 4πr Greek Alphabet Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu f (t) = √ 2π −∞ √ s f (z) −∞ ∞ −∞ ∞ f (t)eiωt dt, Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega N O P T ϒ X ν ξ o π ρ σ τ υ φ χ ψ ω ρ(r )d3r , |r − r | d3 k eik·r k2 + m2 (2π )3 ... Az By Bz Ax Ay A · (B × C) = Bx Cx By A? ?B = Cy xˆ − Ax Az Bx Bz yˆ + Az Ay Bz = C x By Cz A × (B × C) = B A · C − C A · B, Az Bz A · B = Ax Bx + Ay By + Az Bz Ax Ay Bx By − Cy zˆ Ax Az Bx Bz... number) 1 1 B1 = − , B2 = , B4 = B8 = − , B6 = , (Bernoulli numbers) 30 42 www.elsolucionario.net Essential Mathematical Methods for Physicists www.elsolucionario.net www.elsolucionario.net Essential. .. B from the special case as A · B = A · (Bx xˆ + Byyˆ + Bzzˆ ), = Bx A · xˆ + ByA · yˆ + BzA · zˆ , applying Eqs (1.9) and (1.10) = Bx Ax + By Ay + Bz Az, upon substituting Eq (1.8) Hence, A? ?B? ??