Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXwave equation, 714–716, 737, 739series, 115–141convergence of, see convergence of infiniteseriesdifferentiation of, 131finite and infinite, 116integration of, 131multiplication by a scalar, 131multiplication of (Cauchy product), 131notation, 116operations, 131summation, see summation of seriesseries, particulararithmetic, 117arithmetico-geometric, 118Fourier, see Fourier seriesgeometric, 117Maclaurin, 138, 140power, see power seriespowers of natural numbers, 121Riemann zeta, 128, 129Taylor, see Taylor seriesseries solutions of differential equations, 531–550about ordinary points, 535–538about regular singular points, 538–540Frobenius series, 539convergence, 536indicial equation, 540linear independence, 540polynomial solutions, 538, 548–550recurrence relation, 536, 538second solution, 537, 544–548derivative method, 545–548Wronskian method, 544, 580shortest path, 778and geodesics, 976, 982similarity transformations, 283–285, 929, 1092properties of matrix under, 284unitary transformations, 285simple harmonic oscillator, 555energy levels of, 642, 902equation, 535, 566operator formalism, 667simple poles, 838Simpson’s rule, 1004simultaneous linear equations, 292–307and intersection of planes, 300homogeneous and inhomogeneous, 293singular value decomposition, 301–307solution usingCramer’s rule, 299inverse matrix, 295numerical methods, see numerical methodsfor simultaneous linear equationssine, sin(x)in terms of exponential functions, 102Maclaurin series for, 140orthogonality relations, 417singular and non-singularintegral equations, 805linear operators, 249matrices, 263singular integrals, see improper integralssingular point (singularity), 826, 837–839essential, 838, 856removable, 838singular points of ODE, 533irregular, 534particular equations, 535regular, 534singular solution of ODE, 469, 481, 482, 484singular value decompositionand simultaneous linear equations, 301–307singular values, 302sinh, hyperbolic sine, 102, 833, see alsohyperbolic functionsskew-symmetric matrices, 270skewness, 1150, 1227Snell’s law, 788soap films, 780solenoidal vectors, 352, 389solid angleas surface integral, 395subtended by rectangle, 411solid: mass, centre of mass and centroid,193–195source density, 679space curves, 340–344arc length, 341binormal, 342curvature, 342Frenet–Serret formulae, 343parametric equations, 340principal normal, 342radius of curvature, 342radius of torsion, 343tangent vector, 342torsion, 342spaces, see vector spacesspan of a set of vectors, 242sphere, vector equation of, 228spherical Bessel functions, 615, 741of first kind j l (z), 615of second kind n l (z), 615spherical harmonics Yl m (θ, φ), 593–595addition theorem, 594spherical polar coordinates, 361–363area element, 362basis vectors, 362length element, 362vector operators, 361–363volume element, 205, 362spur of a matrix, see trace of a matrixsquare matrices, 249square, symmetries of, 1100square-wave, Fourier series for, 418stagnation points of fluid flow, 873standard deviation σ, 1146of sample, 1224standing waves, 6931329