INDEX Absolute maximum, 14 Absolute minimum, 14 Absolutely integrable, 591 Action, 653 Action at a distance, 109 Addition formula Bessel functions, 537 Alternating series, 313 Amplitude spectrum, 609 Analytic functions, 349 derivative, 384 Taylor series, 11 Antiderivative pirimitive, 36 Arc length, 83 Area of a surface, 173 Argument, 335 function, 3 Associated Laguerre polynomials, Average function, 35 Baker-Hausdorf formula, 294 Basis states, 754 Basis vectors, 141, 167, 245 Bayes criteria, 738 Bayes formula, 675 Bell states entanglement, 771 Bernoulli equation, 423 Bessel function addition formula, 537 Jacobi-Agner expansion, 537 Bessel functions boundary conditions, 531 expansion theorem, 531 first-kind, 513 generating functions, 519 integral definitions, 521 orthogonality roots, 527 recursion relations, 518 second-kind, 514 third-kind, 517 566 Weber integral, 533 Wronskians, 522 Bessel s equation series solution, 510 Bessel s inequality, 590 Binomial coefficients, 681 Binomial distribution, 701 moments, 707 Essentials of Mathematical Methods in Science and Engineering. By 5. SelGuk Bayin Copyright @ 2008 John Wiley & Sons, Inc. 793
794 INDEX Binomial formula binomial coefficients, 323 Bit, 723 Boltzmann distribution gases, 686 solids, 684 Bose-Einstein condensation, 688 Bose-Einstein distribution, 687 Boundary conditions, 4 spherical coordinates, 558 Boundary point, 2 Bounded variation, 594 Bra-ket vectors, 297 Brachistochrone problem, 664 Buffon s needle, 677 Cartesian coordinates, 62 Cartesian tensors, 148 Cauchy criteria, 306 Cauchy integral formula, 382 Cauchy principal value, 41 Cauchy product, 316 Cauchy-Euler equation explicit solutions, 441 Cauchy-Goursat theorem, 376 Cauchy-Riemann conditions, 346 polar coordinates, 351 Cauchy-Schwartz inequality, 586 Cbit, 746, 767 operations, 750 Central moment, 705 Change of basis, 254 Channel blocker, 765 Characteristic equation, 258 Characteristic value eigenvalue, 257 Chebyshev s theorem, 710 Chi-square, 700 Clairaut equation, 428 Closed set, 2 Collectively independent events, 675 Combinations, 681 Commutator, 756 Comparison test, 309 Completeness, 274, 754 Complex algebra, 332 Complex conjugate, 334 Complex correlation function modified, 615 Complex functions exponentials, 354 hyperpolic functions, 357 inverse trigonometric functions, 362 limits and continuity, 344 logarithmic function, 358 polynomials, 354 powers, 359 trigonometric functions, 356 Complex infinity, 339 Complex integrals contour integrals, 370 indefinite integrals, 379 Complex plane extended, 339 Complex series convergence, 393 Laurent series, 389 Maclaurin series, 388 Taylor series, 385 Components, 275 covariant /contravariant, 159 Compressible flow, 81 Conditional probability, 673 Conjugate harmonic functions, 351 Conjugate variables, 753 Conservative forces, 118 Constraints, 659 Continuity, 4 piecewise, 596 Continuity equation, 129 Contour integrals, 370, 373 Contraction tensors, 150 Contravariant components, 159 Control bit, 770 Convergence absolute, 309 conditional, 309 integrals conditionally convergent, 37 series, 309 uniform, 309 Convergence tests, 309 Convolution, 621 Coordinate axes, 155 Coordinate curves, 155 Coordinate surfaces, 155 Coordinates components, 246 Correlation coefficient, 610 modified, 611 Correlation function, 610 Coulomb gauge, 126 Covariant components, 159 Cramer s rule, 226 Critical point, 16 Cross product
INDEX 795 vector product, 61 Cryptograph, 775 Cryptography, 775 Cumulative distribution, 697 Curl, 77 Curl-meter, 105 Curvilinear coordinates, 154 Cylindrical coordinates, 187, 191 Darboux sum, 33 De Broglie relation, 753 Decision theory, 735 Decoherence, 769, 775 Del operator gradient, 74 DeMoivre s formula, 336 Dense coding, 776 Dependent variable function, 3 Derivatives chain rule, 22 Determinants, 220 Laplace development, 222 minor, 221 order, 221 properties, 223 rank, 222 Differential equations exact equations integrating factors, 442 explicit solutions, 408 first-order, 410 exact, 417 integrating factors, 419 linear, 416 methods of solutions, 412 F robenius method, 452 first-order equations, 462 general solution, 408 harmonic oscillator, 435 homogeneous nonhomogeneous, 409 implicit solution, 408 initial conditions, 409, 452 linear and higher order, 450 operator approach, 437 particular solution, 408, 444 quadratures, 408 second-order, 429 methods of solution, 430 singular solution, 408 uniqueness of solutions, 452 Differential operators differential equations, 409 Diffusion equation Cartesian coordinates, 550 cylindrical coordinates, 572 heat flow equation, 541 spherical coordinates, 564 Dirac s bra-ket vectors, 297 Dirac-delta function, 618, 699 Direction cosines, 142 Directional derivative, 75 Dirichlet boundary condition, 558 Discrete distributions, 700 binomial, 701 Poisson, 703 uniform, 701 Displacement vector, 168 Distribution function, 694 Distribution functions arcsine, 694 Cauchy, 717 chi-square, 700 double triangle, 717 exponential, 699 gamma, 699 Gaussian, 699 hypergeometric, 719 Polya, 718 probability theory, 696 Rayleigh, 718 uniform, 698 Distributions expected value, 705 mean, 705 standart deviation, 705 variance, 705 Divergence div operator, 77 integral definition, 82 Divergence theorem Gauss s theorem, 82 Domain function, 3 Domain of definition, 343 Dominant strategies, 781 Double integrals, 47 properties, 49 Dual spaces, 297 Duality, 753 Dummy index tensors, 151 Duplication formula, 521 Eigenstates, 754 Eigenvalue characteristic value, 257
796 INDEX degenerate, 257 Eigenvalue problem symmetric matrices, 277 degenerate roots, 278 distinct roots, 278 Eigenvectors, 258 Electrostatics, 128 Entanglement Bell states, 771 Entire function, 349 Entropy solids, 689 Entropy of information, 729 Equation of continuity, 81 Equilibrium, 16 Essential singular point, 394 Euler constant, 516 Euler equation alternate form, 642 variational analysis, 642 Euler s formula, 354 Events certain, 667 collectively independent, 675 impossible, 667 independent, 674 mutually exclusive, 669 pairwise independent, 675 random, 667 Exact differentials path independence, 114 Expectation value, 755 Expected gain, 739 Expected loss, 739 Expected value, 705 Extensive forms, 739 Extremum local absolute, 15 maximum minimum, 15 with conditions, 18 Extremum points, 637 Fermi-Dirac distribution, 689 Fields, 242 Flip operator, 766 Fourier series change of interval, 602 convergence, 593 differentiation, 603 Dirichlet integral, 594 exponential form, 592 fundamental theorem, 596 generalized, 588 Gibbs phenomenon, 598 integral representation, 594 integration, 603 periodic extention, 600 Riemann localization theorem, 595 sine/cosine series, 602 square wave, 597 triangular wave, 599 trigonometric, 591 uniqueness, 597 Fourier transform correlation function, 615 derivative, 621 existence, 620 inverse, 615 properties, 621 Free index tensors, 151 Frequency of occurrence, 672 Frequency spectrum, 609, 617 Frobenius method, 452 Function, 2 Functionals, 638 Fundamental theorem averages, 704 calculus, 36 Game theory, 737 Gamma distribution, 699 Gamma function, 458, 521, 526, 536 duplication formula, 526, 527 Gates, 747, 752, 758 Gauss s law, 118 Gauss s method linear equations, 217 Gauss s theorem divergence theorem, 82 Gauss-Jordan reduction, 218 Gaussian distribution, 699 moments, 706 Gaussian surface, 111 General boundary condition, 558 General solution, 409 Generalized coordinates, 154, 653 area element, 171 curl, 185 divergence, 182 gradient, 179 Laplacian, 186 orthogonal, 186 volume element, 177 Geometric probability, 677 Geometric series, 310
INDEX 797 Gibbs phenomenon, 598 Gradient del operator, 74 generalized coordinates, 179 Gram-Schmidt orthogonalization, 276 Gramian, 275 Gravitational field, 108 Birkhoff s theorem, 112 stars, 111 Gravitational potential, 116 Gravitational potential energy uniform sphere, 121 Green s first identity, 107 Green s second identity, 107, 137 Green s theorem, 91 Cauchy-Goursat theorem, 376 multiply connected domains, 96 Hadamart operator, 766 Hamilton s principle, 651 Hamiltonian operator, 758 Hankel functions, 517 Harmonic functions, 350 Harmonic series, 310 Heat flow equation Cartesian coordinates, 550 cylindrical coordinates, 572 spherical coordinates, 564 Heisenberg uncertainty, 753 Helmholtz spherical coordinates, 563 Helmholtz equation, 542 cylindrical coordinates, 570 Helmholtz theorem, 122 Hermite equation series solution, 487 Hermite polynomials, 491 contour integral definition, 492 generating function, 494 Hermite series, 499 orthogonality, 496 recursion relations, 495 Rodriguez formula, 493 special values, 495 weight function, 497 Hermitian, 289 Hermitian operators, 294 Hilbert space, 296, 754 completeness, 754 orthogonality, 754 Homogeneous differential equation, 409 Identity matrix unit matrix, 209 Identity operator, 766 Identity tensor, 152 Implicit functions, 25 Implicit solution, 408 Improper transformations, 140 Impulse function Dirac-delta function, 618 Incompressible fluids, 129 Independent variable function, 3 Indicia1 equation, 453 Inflection point, 15 Information conditional probabilities, 733 continuous distributions, 733 H-function, 729 joint events, 732 unit, 723 Information content, 728 Information processing, 726 Information value, 728 Initial conditions boundary conditions, 409 Inner product, 272, 586 norm, 586 Inner product space, 274 Integral indefinite, 36 Integral test, 309 Integrals absolutely convergent conditionally convergent, 37 Cauchy principal value, 41 Darboux sum, 33 double triple, 47 improper, 37 M-test, 42 multiple, 50 with a parameter, 42 Integrating factor, 419 Integration by parts, 37 Integration constant, 409 Interference, 757 Interferometer Mach-Zehnder, 760 Invariants, 147, 178 Inverse basis vectors, 167 Inverse Fourier transform, 615 Inverse functions, 30 Inverse matrix, 230 Inverse transformation, 144
798 INDEX Irrotational flow, 129 Isolated singular points, 394 Jacobi determinant implicit functions, 27 Jacobi identity, 130 Jacobi-Agner expansion Bessel function, 537 Jacobian, 157 inverse functions, 30 Jordan arc, 372 Kinetic energy, 87 Kronecker delta, 63 identity tensor, 152 L 'HBpit a1 's rule limits, 6 Lagrange multiplier extremum problems, 20 Lagrange's equation, 426 Lagrangian, 653, 657 constraints, 659 Laguerre equation, 500 series solution, 500 Laguerre polynomials, 502 contour integral definition, 502 generating function, 504 Laguerre series, 506 orthogonality, 505 Rodriguez formula, 503 special values, 504 Laplace development, 222 Laplace equation, 119, 541, 650 Cartesian coordinates, 546 cylindrical coordinates, 569 spherical coordinates, 557 Laplace transform, 622 differential equation, 625 inverse, 623 transfer functions, 627 Laplacian, 105 Laurent series, 389 Law of large numbers, 712 ergodic theorems, 705 Left derivative, 6 Legendre equation, 470 polynomial solutions, 474 series solution, 470 Legendre polynomials, 474 generating function, 478 Legendre series, 484 orthogonality, 482 recursion formulas, 481 Rodriguez formula, 477 special values, 480 Leibnitz's rule, 43 Levi-Civita symbol permutation symbol, 152 Limit comparison test, 309 Limits, 5 Line element, 164, 168 Line integrals arc length, 83 Linear combination, 244 Linear equations, 216 homogeneous, 233 Linear independence, 244, 275 Linear spaces vector space, 242 Linear transformations matrix representation, 293 operators, 249 Lines, 68 Liouville theorem, 402 Lorentz gauge, 129 M-test integrals, 42 Mach-Zehnder interferometer, 760 mathematics, 763 Maclaurin series, 11, 324, 388 Magnet ost at ics, 128 Magnitude, 58 Mapping function, 2 Matrices adjoint, 232 algebra, 209 cofactor, 231 diagonal, 209 dimension, 207 Hermitian, 294 self-adjoint, 289 identity matrix, 209 inverse matrix, 230 hear equations, 216 orthogonal, 287 rectangular, 207 row matrix column matrix, 208 spur trace, 211 square order, 208 submatrix partitioned matrix, 215 symmetry, 209
INDEX 799 transpose, 208 unitary, 291 zero matrix null matrix, 209 Maxwell s equations, 128 Mean, 705 function, 35 Mean square error, 590 Mean value theorem Rolle s theorem, 36 Median, 706 Method of elimination, 218 Metric tensor, 165 Minimax criteria, 738 Minkowski inequality, 586 Minor determinants, 221 Mixed state, 755 Modified Bessel functions, 523 Modulus, 334 Moment of inertia scalar, 285 Moment of inertia tensor, 265 Multinomial coefficients, 681 Multiple integrals, 50 Multiple-to-one functions, 3 Multiplication theorem, 673, 674 Multiply connected domain, 381 Multivalued functions, 3 Multivalued functions complex functions, 358 principal value, 358 Mutually exclusive events, 762 Nash equilibrium, 742 Natural boundary conditions, 642 Necker cubes, 748 Neighborhood, 2 Neumann boundary condition, 558 Neumann function, 515 No-cloning theorem, 768 control Qbit target Qbit, 770 Norm, 58 magnitude, 274 Riemann integral, 34 Normal distribution, 699 Normal forms, 738 Novelty value, 728 Null matrix zero matrix, 209 Null set, 2 Numbers scalars, 242 Nyquist sampling frequency, 609 One-time-pad quantum cryptography, 776 Open set, 2 Operators on Cbits, 750 Ordinary derivative, 6 Orthogonal functions completeness, 590 convergence mean, 590 inner product, 586 linear independence, 587 theory, 586 Orthogonal matrices, 287 Orthogonal transformations, 140 Orthogonality, 274, 754 Orthogonality condition, 143 Outer product tensors, 149 Pairwise independent events, 675 Parceval s formula, 590 Parceval s theorem, 622 Partial derivative, 6 Particular solution, 409 Partitioned matrices symmetry, 214 Path independence, 113 Payoff matrix, 738 Permutation symbol, 65 Levi-Civita tensor, 152 Permutations, 681 Phase shift operator, 766 Phase spectrum, 609 Piecewise continuous, 5 Planck formula, 753 Planes equation, 69 Poisson distribution, 703 moments, 708 Poisson s equation, 119 Poles singular points, 394 Potential energy gravitational, 117 Power series, 321 Primitive antiderivative, 36 Principal coordinates, 265
800 INDEX Principal directions, 265 Principal moments of inertia, 265 Prior uncertainty, 729 Probability classical definition, 668 entropy, 689 Probability amplitudes, 755 Probability density, 754 Probability density function, 698 Probability theory basic theorems, 669 Bayes formula, 675 Buffon s needle, 677 Chebyshev s theorem, 710 combinations, 678 compound element, 669 conditional probability, 673 distribution function, 694 elementary event, 669 event, 669 frequency of occurrence, 672 fundamental theorem, 704 geometric probability, 677 law of large numbers, 705 multiplication theorem, 673, 674 permutations, 678 random variables, 693 sample space, 668 simple event, 669 statistical definition, 672 total probability, 675 Proper transformations, 140 Protocols quantum cryptography, 776 Pseudotensors, 178 Cartesian tensors, 153 Pure state. 756 Qbit, 767 Qbit operators, 766 Qbit versus Cbit, 767 Quadratic forms, 285 Quadratures differential equations, 408 Quantum cryptography protocols, 776 Quantum dense coding, 776 Quantum information cryptography, 775 Vernam coding, 775 Quantum mechanics, 752 Radius of convergence, 322 Random variables, 693 Range, 343 function, 3 Rank, 222 tensors, 147 Ratio test, 310 Residue theorem, 398 Riccati equation, 424 Riemann integral, 34 Riemann localization theorem, 595 Riemann sphere, 340 Riemann-Lebesgue lemma, 591 Right derivative, 6 Rolle s theorem mean value theorem, 36 Root test, 310 Row matrix, 208 Sample space, 668 Sampling property, 618 Sampling theorem Shannon, 609 Scalar field, 71 Scalar product dot product inner product, 60 Schrodinger equation time-dependent spherical coordinates, 566 time-independent spherical coordinates, 565 Schwarz inequality, 37, 65 Selenoidal fields, 82 Self-adjoint operators, 294 Self-energy gravitational, 121 Separation of variables, 542 Cartesian coordinates, 542 cylindrical coordinates, 567 spherical coordinates, 553 Sequences Cauchy criteria, 304 upper/lower limit, 307 Series Cauchy product, 316 convergence, 309 grouping, 314 indeterminate forms, 325 infinite, 308 multiplication, 314, 316 rearrangement, 315 Series of functions uniform convergence, 316 Series operations, 314 Signals, 608
INDEX 801 Similarity transformation, 256 Simple closed curve Jordan curve, 372 Simple pole, 394 Simply connected domain, 381 Singular point, 349 Singular points classification, 394 Singular solution, 409 Smooth curve, 372 Smooth functions very smooth functions, 596 Spectrum eigenvalues, 258 Spherical Bessel functions, 525 Spherical coordinates, 193 Spherical pendulum, 666 Spur tensors, 150 trace, 211 Square matrix, 208 Standart deviation, 705 State function wave function, 297, 754 State vector collapse, 755 wave function, 754 Stationary functions, 639 Stationary points, 637 Stationary values, 638 Statistical information, 727 Statistical probability, 672 Stereographic projection, 340 Stirling s approximation, 312 Stokes s theorem, 97, 102 Strain tensor, 153 Streamlines, 71 Stress tensor, 153 Submatrices, 214 Summation convention Einstein, 163 Superposed state, 757 Surface integrals, 88 Tangent plane to a surface, 75 Target bit, 770 Taylor series, 11, 324, 388 radius of convergence, 388 remainder, 387 Teleportation, 777 Teleportee, 777 Temperature, 691 Tensor density, 178 Cartesian tensors, 153 Tensors algebra, 148 rank, 147 spur trace, 150 tensor product outer product, 149 transpose, 149 Total differential, 10 Total probability, 675 Trace spur, 211 tensors, 150 Trace formula, 295 Transfer functions Laplace transforms, 627 Transformation matrix, 143 Transformations active/passive, 286 algebra, 252 inverse, 254 linear, 249 matrix representation, 250 product, 253 similar, 255 unitary, 291 Transpose, 149, 208 Ikaveler s dilemma, 742 Triangle inequality, 65 Triple product, 66 Uncertainty principle, 753 Uniform convergence M-test, 42 properties, 319 Weierstrass M-test, 318 Uniform distribution, 698, 701 Union, 2 Unitary matrices, 291 Unitary space, 274 Unitary transformation, 758 Variational analysis Euler equation, 642 functionals, 638 general case, 647 inverse problem, 650 Laplace equation, 650 minimal surfaces, 649 natural boundary conditions, 642 notation, 645 stationary functions, 639
802 INDEX stationary paths, 638 Vector algebra, 62 Vector field, 71 Vector multiplication, 60 Vector product cross product, 61 Vector spaces, 242 basis vectors, 245 dimension, 246 generators, 244 Vectors addition, 60 differentiation, 72 magnitude norm, 58 vector spaces, 242 Velocity potential, 129 Vernam coding, 775 Wave equation, 541 Cartesian coordinates, 544 cylindrical coordinates, 570 spherical coordinates, 563 Wave function state vector, 754 Weak links, 746 Weber integral Bessel functions, 533 Work done, 84, 113 Wronskian, 429 differential equations, 440 Zero matrix null matrix, 209 Zero-sum games, 738