Applied analytical mathematics for physical scientists:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1975
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 651 S. graph. Darst. |
| ISBN: | 0471189979 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
I Linear Vector Spaces 1
1.0 Introduction 1
1.1 Definition of a linear vector space 6
a. Groups 6
b. Fields 7
c. Linear vector spaces 7
1.2 Inner product 8
1.3 Convergence and complete spaces 10
a. Continuity and uniform continuity of a function 10
b. Convergence of a series of functions 11
c. Cauchy convergence 12
d. Proof of completeness of E» 13
1.4 Linear manifolds and subspaces 16
a. Linear independence 16
b. Linear manifolds 17
1.5 Basis for En and E«, 18
1.6 Schmidt orthogonalization process 19
1.7 Projection theorem 20
1.8 Linear functionals 23
2 Operators on Linear Spaces 30
2.0 Introduction 30
2.1 Matrices and determinants 30
a. Definitions of and basic operations with matrices 30
b. Determinants 33
c. Inverse of a matrix 38
d. Direct product of matrices 39
xiii
xiv | CONTENTS
2.2 Systems of linear algebraic equations 41
a. Cramer s rule 41
b. Singular homogeneous systems 42
c. Singular inhomogeneous systems 43
2.3 Gram determinant 44
a. Test for linear independence 44
b. Hadamard s inequality 47
2.4 Definition of a linear operator 47
2.5 Representation of a linear operator by a matrix 48
2.6 Effect of a linear transformation 53
a. Rotation of coordinate axes and of vectors 53
b. Elements of a matrix in different bases 58
c. Covariant vectors, contravariant vectors, and the reciprocal
basis 60
2.7 Inversion of operators 63
a. Separable operators 63
b. Identity plus an infinitesimal operator 70
2.8 Adjoint of an operator 74
2.9 Existence and uniqueness of the solution of Lx = a 76
2.10 Completely continuous operators 80
A2.1 A polynomial expansion for A(x) = det{Oij+8ijX} 85
A2.2 Exponentiation of the two dimensional infinitesimal spatial
rotations 87
A2.3 Proof of Theorem 2.16 88
A2.4 Proof that 3t= 3l(BX for a bounded, self adjoint linear
operator defined on 3£ 92
Table 2.2 Results on the inversion of separable linear operators 93
3 Spectral Analysis of Linear Operators 101
3.0 Introduction 101
3.1 Invariant manifolds 102
3.2 Characteristic equation of a matrix 106
3.3 Self adjoint matrices and completeness 108
3.4 Quadratic forms 113
a. Minimax principle 114
b. Simultaneous reduction of two quadratic forms 117
3.5 Simultaneous diagonalization of commuting hermitian mat¬
rices 120
3.6 Normal matrices and completeness 122
CONTENTS | xv
3.7 Functions of an operator 123
A3.1 Lagrange undetermined multipliers 125
A3.2 Derivation of extremal properties of Hermitian quadratic
forms using Lagrange undetermined multipliers 127
A3.3 A direct verification of the minimax principle in a real
three dimensional space 129
4 Complete Sets of Functions 139
4.0 Introduction 139
4.1 Criterion for completeness 140
a. Bessel s inequality 140
b. Approximation in the mean 141
4.2 Weierstrass approximation theorem 142
4.3 Examples of complete sets of functions 147
a. Fourier series 147
b. Legendre polynomials 154
4.4 Riemann Lebesgue lemma 158
4.5 Fourier integrals 161
a. A heuristic approach 161
b. Fourier integral theorem 162
A4.1 An alternative proof of the Weierstrass approximation
theorem in one variable 167
A4.2 Two proofs of the Weierstrass approximation theorem in
two variables 168
A4.3 A proof of the Coulomb expansion 172
A4.4 Derivation of some important relations satisfied by the
Legendre polynomials, Pn(x) 173
A4.5 Derivation of some important relations satisfied by the
associated Legendre polynomials, KT(x) 174
Table 4.1 Some relations satisfied by the Legendre polynomials 177
Table 4.2 A short table of Fourier transformations 178
O Integral Equations 183
5.0 Introduction 183
5.1 Volterra equations 184
a. Equations of the first and second kind 184
b. Connection with ordinary differential equations 191
xvi | CONTENTS
5.2 Classification of Fredholm equations 196
5.3 Successive approximations 197
5.4 Degenerate and completely continuous kernels 200
5.5 Fredholm s theorems 203
5.6 Fredholm s resolvent 204
5.7 Weak singularities 206
5.8 Hilbert Schmidt theory 210
A5.1 A derivation of Fubini s method given in Eqs. 5.26 and 5.29 222
A5.2 A derivation of Fredholm s expression for the resolvent,
(x, y; A) 224
n
A5.3 A direct proof that lira £ a^ p,| 0L( ) | p, =0 228
A5.4 An equivalent definition of completely continuous operators
in terms of strong convergence 229
A5.5 Proof that every bounded sequence of vectors in a Hilbert
space has a weakly convergent subsequence 230
6 Calculus of Variations 236
6.0 Introduction 236
6.1 Extremum of an integral with fixed end points 237
a. Euler Lagrange conditions 237
b. Several dependent variables 245
6.2 Variable end points 245
6.3 Isoperimetric problems 246
6.4 Lagrangian field theories 249
6.5 Noether s theorem 251
7 Complex Variables 261
7.0 Introduction 261
7.1 Definition of a holomorphic function 262
7.2 Cauchy Riemann conditions 264
7.3 Cauchy s theorems 266
a. Cauchy Goursat theorem 267
b. Cauchy s integral formula 272
CONTENTS | xvii
7.4 Taylor series 274
7.5 Zeros and singularities 278
7.6 Liouville s theorem 282
7.7 Laurent series 283
7.8 Theory of residues 286
a. Rational algebraic integrands 291
b. Trigonometric integrands 294
7.9 Multiple valued functions 297
a. Branch points and branch cuts 298
b. Riemann sheets 304
c. Integral along a branch cut 307
d. Dispersion representation 311
e. Principal value integrals 313
7.10 Singular integral equations 318
7.11 Analytic continuation 322
a. Power series 324
b. Poisson integral formula 330
c. Dirichlet problem and conformal mapping 332
d. Schwarz principle of reflection 339
7.12 Integral representations 341
a. F(z)—the gamma function 341
b. Method of steepest descent 346
c. Analytic properties of Fourier integrals 356
7.13 Classical functions 361
a. F(a, b |c| z)—the hypergeometric function 361
b. Pn(z)—the Legendre functions 373
c. Jn(z)—the Bessel function 380
A7.1 The solution to a dispersion theory problem 394
A7.2 A justification for the interchange of a double infinite sum
when anm2:0, Vn, m 397
A7.3 Recursion and orthogonality relations for the Jacobi poly¬
nomials, P^(x) 397
A7.4 Ellipse of convergence of the Legendre polynomial
expansion 403
A7.5 Proof of the convergence of a polynomial expansion by
means of conformal mapping 406
A7.6 Confluent hypergeometric function 410
Table 7.1 Elementary properties of some special functions 415
xviii | CONTENTS
8 Second Order Linear
Ordinary Differential Equations 431
and Green s Functions
8.0 Introduction 431
8.1 Ideal functions 433
a. Test functions 433
b. Linear functional 434
c. Derivatives of ideal functions 435
d. Ideal limits 437
e. Derivatives of discontinuous functions 439
8.2 Existence and uniqueness theorems for homogeneous linear
second order ordinary differential equations 443
a. Ordinary points 443
b. Singular points 451
8.3 Sturm Liouville problem for discrete eigenvalues 457
8.4 Linear differential operators 466
a. Domain of a linear differential operator 466
b. Adjoint and hermitian linear differential operators 467
c. Self adjoint second order linear differential operators 468
8.5 Green s functions 470
a. Inverse of a differential operator 472
b. An existence theorem 473
8.6 Various boundary conditions 475
a. Unmixed homogeneous boundary conditions 475
b. Unmixed inhomogeneous boundary conditions 477
c. Case of nontrivial eigenfunction with unmixed boundary
conditions 478
d. The general case 479
8.7 Eigenfunction expansion of the Green s function 484
8.8 A heuristic discussion of Green s functions 485
8.9 Asymptotic behavior of the solutions to a linear differential
equation 487
8.10 The continuous spectrum 490
8.11 Physical applications of Green s functions 498
a. An example 501
b. The scalar Helmholtz equation 505
c. The Schrodinger equation 509
d. The scalar wave equation 511
e. The diffusion or heat equation 516
CONTENTS | xix
A8.1 Proof of the completeness of the Sturm Liouville eigenfunc
tions via the Hilbert Schmidt theorem 519
A8.2 General condition for a second order linear ordinary
differential operator to be self adjoint 521
A8.3 An orthonormal basis for i C 00, °°) 524
A8.4 Explicit proof that (i(i(k) | i|/(k )) = ^(fc) | f (k )) for the con¬
tinuous spectrum 525
Table 8.1 Orthogonal polynomial solutions to the Sturm Liouville
equation 526
Table 8.2 The Sturm Liouville differential equation subject to
self adjoint boundary conditions 528
Table 8.3 Green s functions in the infinite spatial domain for some
partial differential operators 529
9 Group Theory 538
9.0 Introduction 538
9.1 Definitions and elementary theorems 539
a. Definition of an abstract group and examples 539
b. Cayley s theorem 542
c. Lagrange s theorem 542
d. Cosets, conjugate classes, and invariant subgroups 544
e. Homomorphism 550
9.2 Linear representations of groups 550
9.3 Unitary representations 554
9.4 Irreducible representations 555
a. Schur s lemma 555
b. Completeness 559
9.5 Definitions of continuous groups and of Lie groups 570
9.6 Examples of Lie groups 572
a. Orthogonal group in n dimensions, O(n) 572
b. Unitary group in n dimensions, U(n) 572
c. Special (or unimodular) unitary group in n dimensions,
SU(n) 573
d. Complex orthogonal group in four dimensions, M(4) 573
e. Complex unimodular group in two dimensions, C(2) 573
9.7 Infinitesimal generators and group parameters 574
9.8 Structure constants 577
9.9 Casimir operators and the rank of a group 583
9.10 Homomorphism between the proper rotation group O+(3) and
SU(2) 586
xx I CONTENTS
9.11 Irreducible representations of SU(2) 590
a. Spinor representations 590
b. The rotation matrices 593
c. Representations in the space of spherical harmonics 595
9.12 Algebra of the angular momentum operators 596
a. Spectra of J2 and of J3 596
b. Rotation of angular momentum eigenfunctions 599
9.13 Coupling of two angular momenta 603
a. Product basis and the coupled representation 603
b. Clebesch Gordan theorem and selection rules 608
9.14 Integration of rotation group parameters 609
a. Orthogonality of the rotation matrices 611
b. Completeness of the rotation matrices 615
9.15 Tensor operators and the Wigner Eckart theorem 616
A9.1 A calculation of the O+(3) invariant integration density
function in terms of the class parameter 618
A9.2 A direct calculation of the invariant integration density
function for O+(3) 620
Appendix I Elementary Real Analysis 627
Appendix II Lebesgue Integration and Functional Analysis 637
Symbols and Notations 640
Bibliography 641
Index 645
|
| any_adam_object | 1 |
| author | Cushing, James T. |
| author_facet | Cushing, James T. |
| author_role | aut |
| author_sort | Cushing, James T. |
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| building | Verbundindex |
| bvnumber | BV002242547 |
| callnumber-first | Q - Science |
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| ctrlnum | (OCoLC)1288358 (DE-599)BVBBV002242547 |
| dewey-full | 515/.02/45 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.02/45 |
| dewey-search | 515/.02/45 |
| dewey-sort | 3515 12 245 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV002242547 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:42:39Z |
| institution | BVB |
| isbn | 0471189979 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001473845 |
| oclc_num | 1288358 |
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| physical | XX, 651 S. graph. Darst. |
| psigel | TUB-nveb |
| publishDate | 1975 |
| publishDateSearch | 1975 |
| publishDateSort | 1975 |
| publisher | Wiley |
| record_format | marc |
| spelling | Cushing, James T. Verfasser aut Applied analytical mathematics for physical scientists James T. Cushing New York [u.a.] Wiley 1975 XX, 651 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Analyse mathématique Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Graphentheorie (DE-588)4113782-6 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Angewandte Mathematik (DE-588)4142443-8 gnd rswk-swf Naturwissenschaftler (DE-588)4041423-1 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Graphentheorie (DE-588)4113782-6 s DE-604 Funktionentheorie (DE-588)4018935-1 s Funktionalanalysis (DE-588)4018916-8 s Analysis (DE-588)4001865-9 s Mathematik (DE-588)4037944-9 s Naturwissenschaftler (DE-588)4041423-1 s Physik (DE-588)4045956-1 s Angewandte Mathematik (DE-588)4142443-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001473845&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cushing, James T. Applied analytical mathematics for physical scientists Analyse mathématique Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd Graphentheorie (DE-588)4113782-6 gnd Analysis (DE-588)4001865-9 gnd Angewandte Mathematik (DE-588)4142443-8 gnd Naturwissenschaftler (DE-588)4041423-1 gnd Mathematik (DE-588)4037944-9 gnd Physik (DE-588)4045956-1 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4113782-6 (DE-588)4001865-9 (DE-588)4142443-8 (DE-588)4041423-1 (DE-588)4037944-9 (DE-588)4045956-1 (DE-588)4018916-8 |
| title | Applied analytical mathematics for physical scientists |
| title_auth | Applied analytical mathematics for physical scientists |
| title_exact_search | Applied analytical mathematics for physical scientists |
| title_full | Applied analytical mathematics for physical scientists James T. Cushing |
| title_fullStr | Applied analytical mathematics for physical scientists James T. Cushing |
| title_full_unstemmed | Applied analytical mathematics for physical scientists James T. Cushing |
| title_short | Applied analytical mathematics for physical scientists |
| title_sort | applied analytical mathematics for physical scientists |
| topic | Analyse mathématique Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd Graphentheorie (DE-588)4113782-6 gnd Analysis (DE-588)4001865-9 gnd Angewandte Mathematik (DE-588)4142443-8 gnd Naturwissenschaftler (DE-588)4041423-1 gnd Mathematik (DE-588)4037944-9 gnd Physik (DE-588)4045956-1 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Analyse mathématique Mathematical analysis Funktionentheorie Graphentheorie Analysis Angewandte Mathematik Naturwissenschaftler Mathematik Physik Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001473845&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cushingjamest appliedanalyticalmathematicsforphysicalscientists |