Modern mathematical methods for physicists and engineers:
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2000
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| Ausgabe: | 1. publ. |
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| Beschreibung: | XX, 763 S. |
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| 100 | 1 | |a Cantrell, Cyrus D. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Modern mathematical methods for physicists and engineers |c C. D. Cantrell |
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| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2000 | |
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| 650 | 4 | |a Engineering mathematics | |
| 650 | 4 | |a Mathématiques | |
| 650 | 4 | |a Mathematik | |
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| _version_ | 1804127164787851264 |
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| adam_text | SUB GOTTINGEN 211 766 267 MODERN MATHEMATICAL METHODS FOR PHYSICISTS AND
ENGINEERS 2000 B 2093 C. D. CANTRELL CAMBRIDGE UNIVERSITY PRESS CONTENTS
PREFACE PAGE XIX 1 FOUNDATIONS OF COMPUTATION 1 1.1 INTRODUCTION 1 1.2
REPRESENTATIONS OF NUMBERS 2 1.2.1 INTEGERS 3 1.2.2 RATIONAL NUMBERS AND
REAL NUMBERS 14 1.2.3 REPRESENTATION S OF NUMBERS AS TEXT 17 1.2.4
EXERCISES FOR SECTION 1.2 20 1.3 FINITE FLOATING-POINT REPRESENTATIONS
21 1.3.1 SIMPLE CASES 21 1.3.2 PRACTICAL FLOATING-POINT REPRESENTATIONS
25 1.3.3 APPROACHING ZERO OR INFINITY GRACEFULLY 28 1.3.4 EXERCISES FOR
SECTION 1.3 30 1.4 FLOATING-POINT COMPUTATION 31 1.4.1 RELATIVE ERROR;
MACHINE EPSILON 31 1.4.2 ROUNDING 32; 1.4.3 FLOATING-POINT ADDITION AND
SUBTRACTION 35 1.4.4 EXERCISES FOR SECTION 1.4 36 1.5 PROPAGATION OF
ERRORS 37 1.5.1 GENERAL FORMULAS 37 1.5.2 EXAMPLES OF ERROR PROPAGATION
39 1.5.3 ESTIMATES OF THE MEAN AND VARIANCE 41 1.5.4 EXERCISES FOR
SECTION 1.5 43 1.6 BIBLIOGRAPHY AND ENDNOTES 45 1.6.1 BIBLIOGRAPHY 45
1.6.2 ENDNOTES 46 2 SETS AND MAPPINGS 47 2.1 INTRODUCTION 47 2.2 BASIC
DEFINITIONS 49 2.2.1 SETS 49 2.2.2 MAPPINGS 53 2.2.3 AXIOM OF CHOICE 62
2.2.4 CARTESIAN PRODUCTS 62 VII VIII CONTENTS 2.2.5 EQUIVALENCE AND
EQUIVALENCE CLASSES 65 2.2.6 EXERCISES FOR SECTION 2.2 67 2.3 UNION,
INTERSECTION, AND COMPLEMENT 68 2.3.1 UNIONS OF SETS 68 2.3.2
INTERSECTIONS OF SETS 69 2.3.3 RELATIVE COMPLEMENT 70 2.3.4 DE MORGAN S
LAWS 71 2.3.5 EXERCISES FOR SECTION 2.3 71 2.4 INFINITE SETS 72 2.4.1
BASIC PROPERTIES OF INFINITE SETS 72 2.4.2 INDUCTION AND RECURSION 73
2.4.3 COUNTABLE SETS 76 2.4.4 COUNTABLE UNIONS AND INTERSECTIONS 77
2.4.5 UNCOUNTABLE SETS 78 2.4.6 EXERCISES FOR SECTION 2.4 80 2.5 ORDERED
AND PARTIALLY ORDERED SETS 82 2.5.1 PARTIAL ORDERINGS 82 2.5.2
ORDERINGS; UPPER AND LOWER BOUNDS 83 2.5.3 MAXIMAL CHAINS 84 2.5.4
EXERCISES FOR SECTION 2.5 84 2.6 BIBLIOGRAPHY 85 3 EVALUATION OF
FUNCTIONS 86 3.1 INTRODUCTION 86 3.2 SENSITIVITY AND CONDITION NUMBER 86
3.2.1 DEFINITIONS 86 3.2.2 EVALUATION OF POLYNOMIALS 87 3.2.3 MULTIPLE
ROOTS OF POLYNOMIALS 89 3.2.4 EXERCISES FOR SECTION 3.2 91 3.3 RECURSION
AND ITERATION 92 3.3.1 FINDING ROOTS BY BISECTION 92 3.3.2
NEWTON-RAPHSON METHOD 92 3.3.3 EVALUATION OF SERIES 95 3.3.4 EXERCISES
FOR SECTION 3.3 97 3.4 INTRODUCTION TO NUMERICAL INTEGRATION 99 3.4.1
RECTANGLE RULES 101 3.4.2 TRAPEZOIDAL RULE 102 3.4.3 LOCAL AND GLOBAL
ERRORS 102 3.4.4 EXERCISES FOR SECTION 3.4 105 3.5 SOLUTION OF
DIFFERENTIAL EQUATIONS 106 3.5.1 EULER S METHOD 107 3.5.2 TRUNCATION
ERROR OF EULER S METHOD 109 3.5.3 STABILITY ANALYSIS OF EULER S METHOD
111 CONTENTS 112 118 120 121 121 122 122 127 130 136 141 143 143 147 149
152 152 158 162 164 165 165 167 169 171 173 175 175 179 180 181 184 184
184 186 189 189 VECTOR SPACES 191 5.1 INTRODUCTION 191 5.2 BASIC
DEFINITIONS AND EXAMPLES 193 5.2.1 AXIOMS FOR A VECTOR SPACE 193 5.2.2
SELECTED REALIZATIONS OF THE VECTOR-SPACE AXIOMS 194 3.6 3.5.4 3.5.5
SELECTED FINITE-DIFFERENCE METHODS EXERCISES FOR SECTION 3.5
BIBLIOGRAPHY GROUPS, 4.1 4.2 4.3 4.4 4.5 4.6 4.7 RINGS, AND FIELDS
INTRODUCTION GROUP; 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7 4.2.8
4.2.9 3 AXIOMS TWO-ELEMENT GROUP ORBITS AND COSETS CYCLIC GROUPS
DIHEDRAL GROUPS CUBIC GROUPS CONTINUOUS GROUPS CLASSES OF CONJUGATE
ELEMENTS EXERCISES FOR SECTION 4.2 GROUP HOMOMORPHISMS 4.3.1 4.3.2 4.3.3
4.3.4 DEFINITIONS AND BASIC PROPERTIES NORMAL SUBGROUPS DIRECT PRODUCT
GROUPS EXERCISES FOR SECTION 4.3 *SYMMETRIC GROUPS 4.4.1 4.4.2 4.4.3
4.4.4 4.4.5 RINGS 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 FIELDS 4.6.1 4.6.2 4.6.3
PERMUTATIONS CAYLEY S THEOREM CYCLIC PERMUTATIONS EVEN AND ODD
PERMUTATIONS EXERCISES FOR SECTION 4.4 AND INTEGRAL DOMAINS AXIOMS AND
EXAMPLES BASIC PROPERTIES OF RINGS RATIONAL NUMBERS *RING HOMOMORPHISMS
EXERCISES FOR SECTION 4.5 AXIOMS AND EXAMPLES * GALOIS FIELDS EXERCISES
FOR SECTION 4.6 BIBLIOGRAPHY CONTENTS 5.2.3 VECTOR SUBSPACES 201 5.2.4
COMMENTS ON VECTOR-SPACE AXIOMS 205 5.2.5 EXERCISES FOR SECTION 5.2
208 5.3 LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 211 5.3.1 DEFINITIONS
211 5.3.2 BASIC RESULTS ON LINEAR DEPENDENCE 212 5.3.3 EXAMPLES OF
LINEAR INDEPENDENCE 216 5.3.4 EXERCISES FOR SECTION 5.3 219 5.4 BASES
AND DIMENSION 221 5.4.1 DIMENSION OF A VECTOR SPACE 221 5.4.2 SELECTED
REALIZATIONS OF VECTOR-SPACE BASES 225 5.4.3 VECTOR-SPACE ISOMORPHISMS
228 5.4.4 GAUSSIAN ELIMINATION AND LINEAR DEPENDENCE 232 5.4.5 EXERCISES
FOR SECTION 5.4 237 5.5 COMPLEMENTARY SUBSPACES 239 5.5.1 VECTOR
COMPLEMENTS AND DIRECT SUMS 239 5.5.2 DEFINITION OF COMPLEMENTARY
SUBSPACES 240 5.5.3 DIMENSIONS OF COMPLEMENTARY SUBSPACES 241 5.5.4
DIRECT SUMS OF VECTOR SPACES 242 5.5.5 BASES OF COMPLEMENTARY SUBSPACES
243 5.5.6 EXAMPLES OF DIRECT SUMS OF VECTOR SPACES 244 5.5.7 EXERCISES
FOR SECTION 5.5 245 5.6 BIBLIOGRAPHY AND ENDNOTES 246 5.6.1 BIBLIOGRAPHY
246 5.6.2 ENDNOTES 246 LINEAR MAPPINGS I 248 6.1 LINEAR MAPPINGS AND
THEIR MATRICES 248 6.1.1 BASIC PROPERTIES 248 6.1.2 MATRIX OF A LINEAR
MAPPING 250 6.1.3 COMPUTATION OF MATRIX PRODUCTS 261 6.1.4 INVARIANT
SUBSPACES AND DIRECT SUMS 263 6.1.5 OTHER EXAMPLES OF LINEAR MAPPINGS
265 6.1.6 EXERCISES FOR SECTION 6.1 268 6.2 NONSINGULAR LINEAR MAPPINGS
271 6.2.1 DEFINITIONS AND BASIC PROPERTIES 271 6.2.2 CHANGE OF BASIS 275
6.2.3 PERMUTATION MATRICES 277 6.2.4 GENERAL LINEAR GROUP OF A VECTOR
SPACE 278 6.2.5 EXERCISES FOR SECTION 6.2 278 6.3 SINGULAR LINEAR
MAPPINGS 281 6.3.1 SINGULARITY AND LINEAR DEPENDENCE 281 6.3.2
VISUALIZATION OF SINGULAR LINEAR MAPPINGS 282 CONTENTS 6.3.3 NULL SPACE
OF A LINEAR MAPPING 283 6.3.4 OTHER EXAMPLES OF A SINGULAR LINEAR
MAPPINGS 285 6.3.5 EXERCISES FOR SECTION 6.3 287 6.4 INTRODUCTION TO
DIGITAL FILTERS 288 6.4.1 DEFINITIONS 288 6.4.2 NOISE AMPLIFICATION BY
DIGITAL FILTERS 291 6.4.3 DIFFERENCE OPERATORS 292 6.4.4 EXERCISES FOR
SECTION 6.4 298 6.5 TRACE AND DETERMINANT 299 6.5.1 TRACE OF A LINEAR
MAPPING 299 6.5.2 DETERMINANTS 300 6.5.3 EXERCISES FOR SECTION 6.5 309
6.6 SOLUTION OF LINEAR EQUATIONS 310 6.6.1 BASIC FACTS ABOUT LINEAR
EQUATIONS 310 6.6.2 MATRIX FORMULATION OF GAUSSIAN ELIMINATION 312 6.6.3
COMPUTATIONAL ASPECTS OF GAUSSIAN ELIMINATION 317 6.6.4 LU AND LDM R
DECOMPOSITIONS 317 6.6.5 BASES OF THE RANGE AND NULL SPACE 319 6.6.6
RANK-NULLITY THEOREM 321 6.6.7 EXERCISES FOR SECTION 6.6 322 6.7
COMPLEMENTS OF NULL SPACE 324 6.7.1 QUOTIENT SPACE V/NULL [A] 324 6.7.2
ISOMORPHISM OF THE RANGE TO A COMPLEMENT OF THE NULL SPACE 325 6.7.3
RANK-NULLITY THEOREM (AGAIN) 327 6.7.4 RIGHT INVERSES OF A LINEAR
MAPPING 327 6.7.5 EXAMPLES OF RIGHT INVERSES 328 6.7.6 EXERCISE FOR
SECTION 6.7 , 330 6.8 BIBLIOGRAPHY 330 LINEAR FUNCTIONALS 331 7.1
MOTIVATION FOR STUDYING FUNCTIONALS 331 7.2 DUAL SPACES 332 7.2.1
DEFINITIONS 332 7.2.2 RANGE AND NULL SPACE OF A LINEAR FUNCTIONAL 334
7.2.3 EXERCISES FOR SECTION 7.2 335 7.3 COORDINATE FUNCTIONALS 336 7.3.1
DEFINITIONS 336 7.3.2 COORDINATE FUNCTIONALS ON F 337 7.3.3 ISOMORPHISM
OF V* TO V 338 7.3.4 COORDINATE FUNCTIONALS ON TWO-DIMENSIONAL EUCLIDEAN
SPACE 339 7.3.5 COORDINATE FUNCTIONALS AND THE RECIPROCAL LATTICE 341
7.3.6 ISOMORPHISM OF V TO V** 345 7.3.7 EXERCISES FOR SECTION 7.3 346
CONTENTS 7.4 ANNIHILATOR OF A SUBSPACE 347 7.4.1 DEFINITIONS 347 7.4.2
BASES OF THE ANNIHILATOR 347 7.4.3 EXERCISES FOR SECTION 7.4 - * 348 7.5
OTHER REALIZATIONS OF DUAL SPACES 349 7.5.1 DUAL SPACE OF C 349 7.5.2
DUALOFF Z+ 349 7.5.3 BOUNDARY AND INITIAL CONDITIONS FOR DIFFERENTIAL
EQUATIONS 349 7.6 POLYNOMIAL INTERPOLATION 350 7.6.1 LAGRANGIAN
INTERPOLATION 350 7.6.2 EXERCISES FOR SECTION 7.6 352 7.7 TENSORS 352
7.7.1 DEFINITIONS AND BASIC PROPERTIES 353 7.7.2 COMPONENTS OF
SECOND-RANK TENSORS 356 7.7.3 TENSOR PRODUCTS OF VECTORS 358 7.7.4
TENSORS OF RANK M 361 7.7.5 LINEAR MAPPINGS OF TENSORS 362 7.7.6
^EXERCISES FOR SECTION 7.7 365 INNER PRODUCTS AND NORMS 367 8.1
INNER-PRODUCT SPACES 367 8.1.1 DEFINITIONS 367 8.1.2 CANONICAL INNER
PRODUCTS 369 8.1.3 METRIC TENSOR 372 8.1.4 INDEFINITE INNER PRODUCTS 378
8.1.5 ORTHOGONALITY 379 8.1.6 EXERCISES FOR SECTION 8.1 383 8.2 GEOMETRY
OF INNER-PRODUCT SPACE S 384 8.2.1 PYTHAGORAS S THEOREM 384 8.2.2
ORTHONORMAL BASES 392 8.2.3 ORTHOGONAL POLYNOMIALS 397 8.2.4 EXERCISES
FOR SECTION 8.2 403 8.3 PROJECTION METHODS 406 8.3.1 PROJECTION OF A
VECTOR ONTO A SUBSPACE: DEFINITION 406 8.3.2 ORTHOGONAL PROJECTORS 407
8.3.3 ORTHOGONAL COMPLEMENT 409 8.3.4 EXERCISES FOR SECTION 8.3 416 8.4
LEAST-SQUARES APPROXIMATIONS 417 8.4.1 MOTIVATION 417 8.4.2 ABSTRACT
FORMULATION 418 8.4.3 INEQUALITIES FOR LEAST-SQUARES APPROXIMATIONS 419
8.4.4 APPROXIMATION BY FINITE FOURIER SUMS 420 8.4.5 CHEBYSHEV
APPROXIMATIONS 421 CONTENTS 8.4.6 MAPPING A FUNCTION TO ITS FOURIER
COEFFICIENTS 422 8.4.7 EXERCISES FOR SECTION 8.4 423 8.5 DISCRETE
FOURIER TRANSFORM 424 8.5.1 APPROXIMATION OF FOURIER COEFFICIENTS 424
8.5.2 DISCRETE FOURIER BASIS 425 8.5.3 PERIODIC EXTENSION 428 8.5.4
ALIASING 430 8.5.5 SAMPLING THEOREM AND ALIAS MAPPING 433 8.5.6 EXERCISE
FOR SECTION 8.5 437 8.6 VOLUME OF AN M-PARALLELEPIPED 437 8.6.1
PARALLELEPIPEDS 437 8.6.2 RECURSIVE DEFINITION OF VOLUME 438 8.6.3
VOLUME AS A DETERMINANT 438 8.6.4 DETERMINANT AS A VOLUME RATIO 440
8.6.5 JACOBIAN DETERMINANT 441 8.6.6 EXERCISE FOR SECTION 8.6 443 8.7
VECTOR AND MATRIX NORMS 443 8.7.1 VECTOR NORMS 443 8.7.2 NORM OF A
LINEAR MAPPING 446 8.7.3 MATRIX NORMS 450 8.7.4 NORM OF AN INTEGRAL 453
8.7.5 EXERCISES FOR SECTION 8.7 453 8.8 INNER PRODUCTS AND LINEAR
FUNCTIONALS 455 8.8.1 INTRODUCTION 455 8.8.2 INNER-PRODUCT MAPPING 456
8.8.3 INVERSE INNER-PRODUCT MAPPING 459 8.8.4 EXERCISES FOR SECTION 8.8
. 464 8.9 BIBLIOGRAPHY AND ENDNOTES 464 8.9.1 BIBLIOGRAPHY 465 8.9.2
ENDNOTES 465 LINEAR MAPPINGS II 466 9.1 DYADS 466 9.1.1 MOTIVATION 466
9.1.2 DEFINITION OF A DYAD 466 9.1.3 DYADIC EXPANSIONS 469 9.1.4
RESOLUTIONS OF THE IDENTITY MAPPING 471 9.1.5 EXERCISE FOR SECTION 9.1
473 9.2 TRANSPOSE AND ADJOINT 473 9.2.1 TRANSPOSE 473 9.2.2 ADJOINT 476
9.2.3 OTHER REALIZATIONS OF THE ADJOINT 480 9.2.4 PROPERTIES OF THE
ADJOINT 483 XIV CONTENTS 9.2.5 HERMITIAN AND SELF-ADJOINT MAPPINGS 485
9.2.6 ISOMETRIC AND UNITARY MAPPINGS 487 9.2.7 EXERCISES FOR SECTION 9.2
491 9.3 EIGENVALUES AND EIGENVECTORS 493 9.3.1 SECULAR EQUATION 493
9.3.2 DIAGONALIZATION OF HERMITIAN MATRICES 495 9.3.3 NORMAL LINEAR
MAPPINGS 502 9.3.4 EXERCISES FOR SECTION 9.3 504 9.4 SINGULAR-VALUE
DECOMPOSITION 507 9.4.1 DERIVATION OF THE SINGULAR-VALUE DECOMPOSITION
507 9.4.2 MATRIX VERSION OF THE SINGULAR-VALUE DECOMPOSITION 509 9.4.3
THE FUNDAMENTAL SUBSPACES OF A LINEAR MAPPING 511 9.4.4 INVERSE AND
PSEUDO-INVERSE IN THE SVD 512 9.4.5 DATA COMPRESSION USING THE SVD 514
9.4.6 EXERCISES FOR SECTION 9.4 514 9.5 LINEAR EQUATIONS II 515 9.5.1
NUMERICAL VERSUS ANALYTICAL METHODS 515 9.5.2 DIAGONAL DOMINANCE 516
9.5.3 CONDITION NUMBER OF THE LINEAR-EQUATION PROBLEM 517 9.5.4 THE LDL
F AND CHOLESKY DECOMPOSITIONS 520 9.6 SELECTED APPLICATIONS OF LINEAR
EQUATIONS 521 9.6.1 THE LINEAR LEAST-SQUARES PROBLEM 521 9.6.2 LINEAR
DIFFERENCE EQUATIONS 523 9.6.3 SOLUTION OF TRIDIAGONAL SYSTEMS 529 9.6.4
EXERCISES FOR SECTION 9.6 530 9.7 BIBLIOGRAPHY 531 10 CONVERGENCE IN
NORMED VECTOR SPACES 532 10.1 METRICS AND NORMS 532 10.1.1 METRIC SPACES
532 10.1.2 NORMED VECTOR SPACES 534 10.1.3 EXAMPLES OF METRIC AND NORMED
VECTOR SPACES 536 10.1.4 OPEN SETS 539 10.1.5 EXERCISES FOR SECTION 10.1
541 10.2 LIMIT POINTS 543 10.2.1 LIMIT POINTS AND CLOSED SETS 543 10.2.2
DENSE SETS AND SEPARABLE SPACES 548 10.2.3 EXERCISES FOR SECTION 10.2
552 10.3 CONVERGENCE OF SEQUENCES AND SERIES 553 10.3.1 CONVERGENCE OF
SEQUENCES 553 10.3.2 NUMERICAL SEQUENCES 558 10.3.3 NUMERICAL SERIES 560
10.3.4 EXERCISES FOR SECTION 10.3 565 CONTENTS XV 10.4 STRONG AND
POINTWISE CONVERGENCE 566 10.4.1 STRONG CONVERGENCE 566 10.4.2 OPERATORS
570 10.4.3 SEQUENCES OF REAL-VALUED FUNCTIONS 572 10.4.4 SERIES OF
REAL-VALUED FUNCTIONS 575 10.4.5 EXERCISES FOR SECTION 10.4 576 10.5
CONTINUITY 577 10.5.1 POINTWISE CONTINUITY 577 10.5.2 UNIFORM CONTINUITY
580 10.6 BEST APPROXIMATIONS IN THE MAXIMUM AND SUPREMUM NORMS 581
10.6.1 BEST APPROXIMATIONS IN THE MAXIMUM NORM 583 10.6.2 BEST
APPROXIMATIONS IN THE SUPREMUM NORM 589 10.6.3 EXERCISES FOR SECTION
10.6 592 10.7 HILBERT AND BANACH SPACES 594 10.7.1 SURVEY OF COMPLETE
METRIC VECTOR SPACES 594 10.7.2 COMPLETE ORTHONORMAL SETS 597 10.7.3
ORTHOGONAL SERIES 599 10.7.4 PRACTICAL ASPECTS OF FOURIER SERIES 603
10.7.5 ORTHOGONAL-POLYNOMIAL EXPANSIONS 610 10.7.6 EXERCISES FOR SECTION
10.7 613 10.8 BIBLIOGRAPHY 615 11 GROUP REPRESENTATIONS 616 11.1
PRELIMINARIES 616 11.1.1 BACKGROUND 616 11.1.2 SYMMETRY-ADAPTED
FUNCTIONS 618 11.1.3 PARTNER FUNCTIONS : 619 11.1.4 EXERCISES FOR
SECTION 11.1 621 11.2 REDUCIBILITY OF REPRESENTATIONS 621 11.2.1
INVARIANT SUBSPACES AND IRREDUCIBILITY 622 11.2.2 SCHUR S LEMMA 623
11.2.3 EIGENVECTORS OF INVARIANT OPERATORS 627 11.2.4 EXERCISES FOR
SECTION 11.2 629 11.3 UNITARITY AND ORTHOGONALITY 630 11.3.1
CONSEQUENCES OF THE REARRANGEMENT THEOREM 630 11.3.2 UNITARY
REPRESENTATIONS 631 11.3.3 ORTHOGONALITY THEOREMS 633 11.3.4 PRODUCT
RELATION FOR CHARACTERS 638 11.3.5 REDUCTION OF UNITARY REPRESENTATIONS
640 11.3.6 CONSTRUCTION OF CHARACTER TABLES 643 11.3.7 CHARACTERS OF
KRONECKER PRODUCTS 644 11.3.8 EXERCISES FOR SECTION 11.3 646 CONTENTS
11.4 TWO-DIMENSIONAL ROTATION GROUP 647 11.4.1 REPRESENTATION SPACE FOR
S 0 (2) 648 11.4.2 REPRESENTATIONS OF S O (2) 649 11.4.3 COMPLETENESS
RELATION FOR {E~ IMB } 650 11.4.4 EXERCISE FOR SECTION 11.4 652 11.5
SYMMETRY AND THE ONE-DIMENSIONAL WAVE EQUATION 652 11.5.1 BOUNDARY
CONDITIONS AND SYMMETRY 652 11.5.2 WAVE EQUATION FOR A VIBRATING STRING
653 11.5.3 BOUNDARY CONDITIONS FOR THE ONE-DIMENSIONAL WAVE EQUATION 653
11.5.4 FORM INVARIANCE OF THE WAVE EQUATION 653 11.5.5 INVARIANCE OF THE
WAVE EQUATION UNDER TRANSLATIONS 656 11.5.6 INVARIANCE OF THE WAVE
EQUATION UNDER LORENTZ TRANSFORMATIONS 656 11.5.7 D ALEMBERT S SOLUTION
OF THE WAVE EQUATION 657 11.5.8 SOLUTION FOR A STRING OF INFINITE LENGTH
658 11.5.9 SOLUTION FOR A STRING OF FINITE LENGTH 659 11.5.10 EXERCISES
FOR SECTION 11.5 661 11.6 DISCRETE TRANSLATION GROUPS 662 11.6.1
MOTIVATION 662 11.6.2 INVARIANCE UNDER THE DISCRETE TRANSLATION GROUP
662 11.6.3 DISCRETE-SHIFT-INVARIANT DIGITAL FILTERS 663 11.6.4
REPRESENTATIONS OF THE DISCRETE TRANSLATION GROUP 664 11.6.5
DISCRETE-TIME TRANSFER FUNCTION 665 11.6.6 EXERCISES FOR SECTION 11.6
668 11.7 CONTINUOUS TRANSLATION GROUPS 669 11.7.1 TRANSLATION GROUP OF
THE REAL LINE 669 11.7.2 IRREDUCIBLE REPRESENTATIONS OF T(R) 669 11.7.3
{VOC} AS MOMENTUM EIGENFUNCTIONS 671 11.7.4 REPRESENTATION OF T(E 2 )
CARRIED BY F K 672 11.7.5 TRANSLATION GROUP OF EUCLIDEAN N-SPACE 673
11.7.6 EXERCISES FOR SECTION 11.7 676 11.8 FOURIER TRANSFORMS 676 11.8.1
FOURIER TRANSFORM IN ONE DIMENSION 676 11.8.2 COMPLETENESS RELATION FOR
THE {F K } 611 11.8.3 FOURIER TRANSFORMS IN N-DIMENSIONAL EUCLIDEAN
SPACE 678 11.8.4 POISSON SUM FORMULA 679 11.8.5 EXERCISES FOR SECTION
11.8 679 11.9 LINEAR, SHIFT-INVARIANT SYSTEMS 680 11.9.1
CONTINUOUS-TIME-SHIFT INVARIANCE 680 11.9.2 CONTINUOUS-TIME TRANSFER
FUNCTION 681 11.9.3 EXERCISES FOR SECTION 11.9 682 CONTENTS XVII 11.10
TWO-DIMENSIONAL EUCLIDEAN GROUP E(2) 682 11.10.1 REPRESENTATION OF E(2)
CARRIED BY {X/F K } 684 11.10.2 DISCUSSION 684 11.10.3 EXERCISES FOR
SECTION 11.10 685 11.11 BIBLIOGRAPHY 685 12 SPECIAL FUNCTIONS 686 12.1
GROUP THEORY AND SPECIAL FUNCTIONS 686 12.1.1 SEPARATION OF VARIABLES
686 12.1.2 SPECIAL FUNCTIONS AS MATRIX ELEMENTS 687 12.1.3 SYMMETRIES OF
THE HELMHOLTZ EQUATION 687 12.1.4 EXERCISE FOR SECTION 12.1 690 12.2
DEFINITION OF THE BESSEL FUNCTIONS 690 12.2.1 FOURIER EXPANSION OF A
PLANE WAVE 690 12.2.2 DEFINITION OF THE BESSEL FUNCTION J M OF INTEGER
ORDER 692 12.2.3 JACOBI-ANGER EXPANSION 692 12.2.4 FREQUENCY CONTENT OF
AN FM SIGNAL 694 12.2.5 EXERCISES FOR SECTION 12.2 694 12.3
BESSEL-FUNCTION ADDITION FORMULAS 695 12.3.1 EXERCISES FOR SECTION 12.3
698 12.4 BESSEL RAISING AND LOWERING OPERATORS 698 12.4.1 RECURRENCE
RELATIONS 698 12.4.2 RAISING AND LOWERING OPERATORS FOR J M 700 12.4.3
RAISING AND LOWERING OPERATORS FOR THE HELMHOLTZ FUNCTIONS 700 12.4.4
EXERCISES FOR SECTION 12.4 701 12.5 BESSEL DIFFERENTIAL EQUATIONS 701
12.5.1 BESSEL S DIFFERENTIAL EQUATION 701 12.5.2 HELMHOLTZ EQUATION IN
TWO DIMENSIONS 702 12.5.3 QUALITATIVE BEHAVIOR OF J M 702 12.5.4
EXERCISES FOR SECTION 12.5 704 12.6 ORTHOGONAL SERIES IN /* 704 12.6.1
BOUNDARY CONDITIONS THAT ENSURE SELF-ADJOINTNESS 704 12.6.2
ORTHOGONALITY RELATIONS 708 12.6.3 FOURIER-BESSEL SERIES 709 12.6.4
EXERCISE FOR SECTION 12.6 711 12.7 VIBRATIONS OF A DRUMHEAD 711 12.7.1
EXERCISE FOR SECTION 12.7 713 12.8 POWER SERIES FOR J M 713 12.8.1
DERIVATION USING RAISING AND LOWERING OPERATORS 713 12.8.2 PROPERTIES
DEDUCED FROM THE POWER SERIES 715 12.8.3 EXERCISES FOR SECTION 12.8 715
XVIII CONTENTS 12.9 COMPLETENESS RELATIONS USING J N 716 12.10
BIBLIOGRAPHY 718 APPENDIX A INDEX OF NOTATION 719 A.I QUANTIFIERS AND
OTHER LOGICAL SYMBOLS 719 A.2 SETS AND MAPPINGS 719 A.3 VECTOR SPACES,
LINEAR MAPPINGS AND MATRICES 720 A.4 NORMS AND INNER PRODUCTS 721 A.5
FUNCTIONS 721 A.5.1 GENERAL NOTATION FOR FUNCTIONS 721 A.5.2 SPECIAL
FUNCTIONS 722 A.6 PROBABILITY 722 APPENDIX B AFFINE MAPPINGS 723 B.I
AFFINE GROUP OF A VECTOR SPACE 723 R B.2 COORDINATE TRANSFORMATIONS 726
B.2.1 ACTIVE TRANSFORMATIONS 727 B.2.2 PASSIVE TRANSFORMATIONS 727 B.2.3
RELATION BETWEEN ACTIVE AND PASSIVE TRANSFORMATIONS 728 B.3 EXERCISES
728 APPENDIX C PSEUDO-UNITARY SPACES 730 APPENDIX D REMAINDER TERM 733
APPENDIX E BOLZANO-WEIERSTRAB THEOREM 735 E.I THE REAL NUMBERS 735 E.2
FINITE-DIMENSIONAL HILBERT SPACES 736 APPENDIX F WEIERSTRAB
APPROXIMATION THEOREM 738 INDEX 745
|
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| bvnumber | BV012519946 |
| callnumber-first | Q - Science |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 950 |
| ctrlnum | (OCoLC)39147887 (DE-599)BVBBV012519946 |
| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV012519946 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:29:00Z |
| institution | BVB |
| isbn | 0521591805 0521598273 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008499401 |
| oclc_num | 39147887 |
| open_access_boolean | |
| owner | DE-703 DE-92 DE-20 DE-355 DE-BY-UBR DE-29T DE-1050 DE-634 DE-83 DE-526 DE-11 |
| owner_facet | DE-703 DE-92 DE-20 DE-355 DE-BY-UBR DE-29T DE-1050 DE-634 DE-83 DE-526 DE-11 |
| physical | XX, 763 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Cantrell, Cyrus D. Verfasser aut Modern mathematical methods for physicists and engineers C. D. Cantrell 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2000 XX, 763 S. txt rdacontent n rdamedia nc rdacarrier Engineering mathematics Mathématiques Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd rswk-swf Ingenieurwissenschaften (DE-588)4137304-2 gnd rswk-swf Mathematik (DE-588)4037944-9 s Ingenieurwissenschaften (DE-588)4137304-2 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008499401&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cantrell, Cyrus D. Modern mathematical methods for physicists and engineers Engineering mathematics Mathématiques Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd Ingenieurwissenschaften (DE-588)4137304-2 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4137304-2 |
| title | Modern mathematical methods for physicists and engineers |
| title_auth | Modern mathematical methods for physicists and engineers |
| title_exact_search | Modern mathematical methods for physicists and engineers |
| title_full | Modern mathematical methods for physicists and engineers C. D. Cantrell |
| title_fullStr | Modern mathematical methods for physicists and engineers C. D. Cantrell |
| title_full_unstemmed | Modern mathematical methods for physicists and engineers C. D. Cantrell |
| title_short | Modern mathematical methods for physicists and engineers |
| title_sort | modern mathematical methods for physicists and engineers |
| topic | Engineering mathematics Mathématiques Mathematik Mathematics Mathematik (DE-588)4037944-9 gnd Ingenieurwissenschaften (DE-588)4137304-2 gnd |
| topic_facet | Engineering mathematics Mathématiques Mathematik Mathematics Ingenieurwissenschaften |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008499401&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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