Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXnomenclature, 1102non-Abelian, 1052–1056order, 1043, 1081, 1082, 1094, 1097, 1100permutation law, 1047subgroups, see subgroupsgroups, examples1and−1 under multiplication, 1043alternating, 1116complex numbers e iθ , 1048functions, 1055general linear, 1073integers under addition, 1043integers under multiplication (mod N),1049–1051matrices, 1054permutations, 1056–1058quaternion, 1073rotation matrices, 1048symmetries of a square, 1100symmetries of an equilateral triangle, 1047H n (x), see Hermite polynomialsHamilton’s principle, 788Hamiltonian, 796Hankel functions H ν (1) (x), H ν (2) (x), 607Hankel transforms, 459harmonic oscillatorsdamped, 239, 451ground-state energy, 796Schrödinger equation, 796simple, see simple harmonic oscillatorheat flowdiffusion equation, 678, 696, 723in bar, 723, 749, 770in thin sheet, 698Heaviside function, 441relation to Dirac δ-function, 441Heisenberg’s uncertainty principle, 435–437Helmholtz equation, 737–741cylindrical polars, 740plane polars, 738spherical polars, 740–741Helmholtz potential, 177hemisphere, centre of mass and centroid, 195Hermite equation, 535, 624–628as example of Sturm–Liouville equation, 566natural interval, 567Hermite polynomials H n (x), 625as special case of confluent hypergeometricfunction, 634generating function, 627graph of, 625normalisation, 626orthogonality, 626recurrence relations, 628Rodrigues’ formula, 626Hermitian conjugate, 256–258and inner product, 258product rule, 257Hermitian forms, 288–292positive definite and semi-definite, 290stationary properties of eigenvectors, 290Hermitian kernel, 816Hermitian matrices, 271eigenvalues, 276–278reality, 276eigenvectors, 276–278orthogonality, 277Hermitian operators, 559–564and physical variables, 650boundary condition for simple harmonicoscillators, 560eigenfunctionscompleteness, 560, 563orthogonality, 561–563eigenvaluesreality, 561Green’s functions, 568–571importance of, 555, 560in Sturm–Liouville equations, 564properties, 561–564superposition methods, 568–571higher-order differential equations, see ordinarydifferential equationsHilbert spaces, 557–559hit or miss, in Monte Carlo methods, 1014homogeneousboundary conditions, see boundaryconditions, homogeneous andinhomogeneousdifferential equations, 490dimensionally consistent, 475, 521simultaneous linear equations, 293homomorphism, 1059–1061kernel of, 1060representation as, 1083Hooke’s law, 953hydrogen atoms-states, 1144electron wavefunction, 208ground-state energy, 800hydrogen molecule, symmetries of, 1041hyperbolaas section of quadratic surface, 292equation for, 16hyperbolic functions, 102–109, 833calculus of, 106–109definitions, 102, 833graphs, 102identities, 104in equations, 105inverses, 105graphs, 106trigonometric analogies, 102–104hyperbolic PDE, 687, 690hypergeometric distribution, 1173mean and variance, 1173hypergeometric equation, 535, 628–632as example of Sturm–Liouville equation, 566,5671317