Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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Contents
Preface
Note to
the
Reader
Listof Symbols
o Mathematical Preliminaries
0.1 Sets .
0.2 Maps .
0.3 Metric Spaces .
0.4
Cardinality........
0.5 MathematicalInduction
0.6
Problems..........................
I Finite-Dimensional VectorSpaces
1 Vectors
and
Transformations
1.1 VectorSpaces . . . . .
1.2 Inner Product .
1.3 Linear Transformations .
1.4
Algebras.
1.5
Problems..........
2
Operator
Algebra
2.1 Algebra
«s:
(V)
v
xi
xxi
1
1
4
7
10
12
14
17
19
19
23
32
41
44
49
49
xiv
CONTENTS
2.2 Derivatives of Functions of Operators .
2.3 Conjugationof Operators . . . .
2.4 HermitianandUnitary Operators
2.5 ProjectionOperators . . . . . . .
2.6 Operators
in Numerical Analysis
2.7
Problems.............
3 Matrices: OperatorRepresentations
3.1
Matrices..........
3.2 Operationson Matrices
..
. . . .
3.3 OrthonormalBases .
3.4 Changeof Basis and SimilarityTransformation.
3.5 TheDeterminant . .
..
3.6 TheTrace
3.7
Problems......
4 SpectralDecomposition
4.1 Direct Sums . . . . . . . . . .
4.2 InvariantSubspaces . . . . . .
4.3 EigeuvaluesandEigenvectors .
4.4 SpectralDecomposition
4.5 Functionsof Operators
4.6 PolarDecomposition
4.7 RealVectorSpaces
4.8
Problems.......
IT
Infinite-Dimensional Vector Spaces
5 Hilbert Spaces
5.1 The Questionof Convergence .
5.2 The Spaceof Square-IntegrableFunctions
5.3
Problems...................
6 Generalized Functions
6.1 ContinuousIndex
6.2 Generalized Functions .
6.3
Problems........
7 Classical Orthogonal Polynomials
7.1 GeneralProperties . .
7.2
Classification.........
7.3 RecurrenceRelations
....
7.4 Examplesof ClassicalOrthogonalPolynomials.
56
61
63
67
70
76
82
82
87
89
91
93
101
103
109
109
112
114
117
125
129
130
138
143
145
145
150
157
159
159
165
169
172
172
175
176
179
7.5
7.6
7.7
Expansion in Termsof OrthogonalPolynomials
..
GeneratingFunctions . . . . . . . . . . . .
Problems .
CONTENTS
xv
186
190
190
8 FourierAnalysis
8.1 Fourier Series
.....
8.2 The FourierTransform .
8.3
Problems........
III Complex Analysis
196
196
208
220
225
9 Complex Calculus 227
9.1 ComplexFunctions " 227
9.2 AnalyticFunctions. . . . . . . " 228
9.3 ConformalMaps
..
. . . . . . . . . . . 236
9A Integrationof ComplexFunctions . . . . . . . . . . . . . " 241
9.5 Derivativesas Integrals 248
9.6 Taylorand Laurent Series . 252
9.7
Problems............................
263
10 Calculus of Residues
10.1 Residues . . . . . . . . . . . . . . . .
10.2 Classificationof Isolated Singularities
10.3 Evaluationof DefiniteIntegrals . .
..
lOA Problems .
11 Complex Analysis: Advanced Topics
ILl
MeromorphicFunctions . . .
11.2 MultivaluedFunctions . . . . . . . .
11.3 AnalyticContinuation . . . . . . . . .
1104
The Gamma and Beta Functions. . . .
11.5 Method of SteepestDescent . . . . . .
11.6 Problems. . . . . . . . . . . . . . . .
270
270
273
275
290
293
293
295
302
309
312
319
IV DifferentialEquations 325
12 Separation
of
Variables in Spherical Coordinates 327
12.1 PDEs of MathematicalPhysics
..
. . . . . . . . . . . . . . 327
12.2 Separation of theAngular Part of the Laplacian. . . . . .
..
331
12.3 Constructionof Eigenvaluesof
L
2.
. . . . . . . . . . 334
1204
Eigenvectorsof L
2:
Spherical Harmonics . . 338
12.5
Problems..
. . . . . . . . . . . . .
.,
. . . . . . . 346
xvi
CONTENTS
13
Second-Order
Linear
Differential
Equations
348
13.1 General Properties of
ODEs.
. . . . . . . . . . . . 349
13.2 Existence and Uniqneness for First-Order DEs . 350
13.3 General Properties
of
SOLDEs . . . 352
13.4 The Wronskian. . . . . . . . . . . . 355
13.5 Adjoint Differential Operators. . . . 364
13.6 Power-Series Solntions
of
SOLDEs . 367
13.7 SOLDEs with Constant Coefficients 376
13.8 The WKB Method . . . . . . . . . . . . 380
13.9 Numerical Solntions of DEs . . . . . . . 383
13.10
Problems.
. . . . . . . . . . . . . . . . . . . . . . . . . . 394
14 Complex Aualysis
of
SOLDEs 400
14.1 Analytic Properties of Complex DEs 401
14.2 Complex SOLDEs . . . . . . . . . . 404
14.3 Fuchsian Differential Equations . . . 410
14.4 The Hypergeometric Functiou . . . . 413
14.5 Confiuent Hypergeometric Functions 419
14.6
Problems.
. . . . . . . . . . . . . . . . . 426
15
Integral
Transforms
and
Differential
Equations
433
15.1 Integral Representation
of
the Hypergeometric Function . . 434
15.2 Integral Representation
of
the Confiuent Hypergeometric
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
15.3 Integral Representation of Bessel Functions 438
15.4 Asymptotic Behavior of Bessel Functions. 443
15.5
Problems..
. . . . . . . . . . . . . . . . . . . . . . . .
..
445
V Operatorson Hilbert Spaces
16 An
Inlroduction
to
Operator
Theory
16.1 From Abstract to Integral and Differential Operators .
16.2 Bounded Operators in Hilbert Spaces . .
16.3 Spectra
of
Linear Operators . .
16.4 Compact Sets .
16.5 Compact Operators .
16.6 Spectrum
of
Compact Operators
16.7 Spectral Theoremfor Compact Operators .
16.8 Resolvents
16.9
Problems.
. . . . . . . . . . . . . . . . .
17
Integral
Equations
17.1 Classification.
449
451
451
453
457
458
464
467
473
480
485
488
488
CONTENTS
xvii
17.2 FredholmIntegra!Equations . 494
17.3
Problems............
505
18 Sturm-Liouville Systems: Formalism 507
18.1 UnboundedOperatorswith CompactResolvent 507
18.2 Sturm-Liouville Systemsand SOLDEs . . . . 513
18.3 OtherPropertiesof Sturm-Liouville Systems . 517
18.4 Problems. . . . . . . . . . . . . . . . . . . . 522
19 Sturm-Lionville Systems: Examples 524
19.1 Expansionsin Termsof Eigenfunctions . 524
19.2 Separationin CartesianCoordinates. . . 526
19.3 Separationin CylindricalCoordinates. . 535
19.4 Separationin SphericalCoordinates. 540
19.5
Problems.................
545
VI Green's Functions
20 Green's Functions
in
One Dimension
20.1 Calculationof SomeGreen's Functions .
20.2 Formal Considerations . . . . . . . . . .
20.3 Green's Functionsfor SOLDOs . . . . .
20.4 EigenfunctionExpansion of Green's Fnnctions .
20.5
Problems..
. . . . . . . . . . . . . . . . . . . . . . . . . .
21 Multidimensional Green's Functions: Formalism
21.1 Propertiesof Partial DifferentialEquations .
21.2 MultidimensionalGFs and Delta Functions .
21.3 FormalDevelopment
...
21.4 IntegralEquationsand GFs
21.5 PerturbationTheory
21.6 Problems. . . . . . . . . .
22 Multidimensional Green's Functions: Applications
22.1 EllipticEquations . .
22.2 ParabolicEquations . . . . . . . . . . . . . . .
22.3 HyperbolicEquations . . . . . . . . . . . . . .
22.4 The FourierTransformTechnique . . . .
22.5 The EigenfunctionExpansionTechnique
22.6 Problems.
..
..
..
..
..
..
.
...
551
553
r-
554
557
565
577
580
583
584
592
596
600
603
610
613
613
621
626
628
636
641
xviii
CONTENTS
VII Groups and Manifolds
23 Group Theory
23.1
Groups.
23.2 Subgroups . . .
23.3 Group Action .
23.4 The Symmetric Group
s,
23.5 Problems. . . . . . . . .
24 Group Representation Theory
24.1 Definitionsand Examples
24.2 OrthogonalityProperties. . .
24.3 Analysis of Representations . .
24.4 GroupAlgebra . . . . . . .
..
. . . . .
24.5 Relationshipof Characters to Those of a Subgroup .
24.6 IrreducibleBasis Functions . . . . . . .
24.7 TensorProduct of Representations
...
24.8 Representationsof the Symmetric Group
24.9
Problems..
..
..
..
.
..
..
.. ..
25 Algebra of Tensors
25.1 MultilinearMappings
25.2 Symmetriesof Tensors .
25.3 ExteriorAlgebra . . . . .
25.4 hmer ProductRevisited .
25.5 The Hodge Star Operator
25.6 Problems. . . . . . . . .
26 Analysis of Tensors
26.1 DifferentiableManifolds. .
26.2 Curves andTangentVectors .
26.3 Differentialof a Map
...
26.4 TensorFields on Manifolds
26.5 Exterior Calculus
...
26.6 SymplecticGeometry
26.7 Problems. . . . . . . .
VIII Lie Groups and Their Applications
27 Lie Groups and Lie Algebras
27.1 Lie Groupsand Their Algebras . . . . . . .
27.2 An Outlineof Lie Algebra Theory
...
27.3 Representationof Compact Lie Groups . . .
649
651
652
656
663
664
669
673
-J
673
680
685
687
692
695
699
707
723
728
729
736
739
749
756
758
763
763
770
776
780
791
801
808
813
815
815
833
845
27.4 Representationof the GeneralLinear Group .
27.5 Representationof Lie Algebras
27.6 Problems .
CONTENTS
xix
856
859
876
28
DifferentialGeometry 882
28.1 VectorFields and Curvature . . . . . . 883
28.2 RiemannianManifolds. . . . 887
28.3 CovariantDerivative and Geodesics . 897
28.4 Isometriesand KillingVectorFields 908
28.5 GeodesicDeviationand Curvature . 913
28.6 GeneralTheory of Relativity 918
28.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 932
29 Lie Gronpsand DifferentialEquations 936
29.1 Synunetriesof AlgebraicEquations. . . . . . 936
29.2 SynunetryGroupsof DifferentialEquations 941
29.3 The CentralTheorems. . . . . . . 951
29.4 Applicationto Some KnownPDEs 956
29.5 Applicationto ODEs. 964
29.6 Problems. . . . . . . . . . . . . . 970
30 Calcnlusof Variations, Symmetries,and ConservationLaws 973
30.1 The Calculusof Variations. . . . . . . . . . . 973
30.2 SymmetryGroupsof VariationalProblems . . . . . . . . 988
30.3 ConservationLaws and Noether's Theorem. . . . . . . . 992
30.4 Applicationto ClassicalField Theory . 997
30.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . .
..
1000
Bibliography 1003
Index 1007
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J
List
of
Symbols
E,
(11')
z
R
R+
C
l\I
ill
~A
AxB
An
U,
«»
A=B
x .....
f(x)
V
3
[a]
gof
iff
ek(a, b)
Cn(or
JRn)
pClt]
P'[t]
P~[t]
Coo
(al
b)
lIall
"belongs to", ("does not belong to")
Set of integers
Set of real nnmbers
Set of positive real nnmbers
Set of complex nnmbers
Set of nonnegative integers
Set of rational nnmbers
Complement
of
the set A
Set of ordered pairs (a, b) with a E A and b E B
{(aI,
a2,
...
, an)lat E Al
Union, (Intersection)
A is eqnivalent to B
x is mappedto
f(x)
via the map f
for all (valnes of)
There exists (a valne
of)
Eqnivalence class to which a belongs
Composition
of
maps f and g
if
and only
if
Set of functions on (a, b) with continnons derivatives np to orderk
Set of complex (or real) n-tnples
Set of polynomials in
t with complex coefficients
Set of polynomials in
t with real coefficients
Set of polynomials with complex coefficients of degree
n or less
Set of
all complex seqnences
(atl~1
snch that
L:~I
latl
2
<
00
Inner prodnct
of
la) and
Ibl
Norm (length)
of
the vector la)
xxii
LIST
OF
SYMBOLS
£-(V)
[8, T]
Tt
At, orA
l.lE/lV
8(x - xo)
Res[f(zolJ
DE, ODE,
PDE
SOLDE
GL(V)
GL(n,q
SL(n,q
71
®72
AJ\B
AP(V)
Set of endomorphisms (linear operators) on vector space V
Commutator of operators 8 and T
Adjoint (hermitian conjugate) of operatorT
Transpose of matrix
A
Direct sum of vector spaces
1.l
and V
Dirac delta function nonvanishing only at
x = xo
Residue
of
f at point zo
Differential equation, Ordinary DE, Partial DE
Second order linear (ordinary) differential equation
Set
of
all invertible operators on vector space V
Set
of
all n x n complex matrices
of
nonzero determinant
Set
of
all n x n complex matrices
of
unit determinant
Tensor product
of
71
and
72
Exterior (wedge) product
of
skew-symmetric tensors A and B
Set of
all skew-symmetric tensors of type
(p,
0) on V