Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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Coulombpoteotial
chargedistribution, 215
pointcharge,214
definition, 210
distribution, 219
Gaussian,212
higher
dimensions,
213
Fourier-Bessel series,536
Fredhohu, 151
Fredhohu alternative, 496
Fredholmequatiou,488,600
Fredhohu,biography, 496
Friedmannmetric,892
Friedmann,biography, 921
Frobenius,681, 850,874
biography, 711
Frobeniusmethod,
370-376
FuchsianDE,
41()...413
definition, 411
function,5
functionalgebra,42
functionof operator, 53,125-128
derivative, 56
functional, 980
linear,34
functional
derivative,
977
functions
of
trig
functions
integralsof, 281
fundamentalsolution,597-{;00
fundamentaltheoremof algebra,251
futurelight cone,837
G-invariance, 937
g-orthogonal,751
g-orthononruU,7~2
Galois,96, 711, 842,942
biography, 653
gammafunction,181,309-312
definition, 309
gaugeinvariance,797
Gauss,96, 183,233,422,463,478,614,
738,797,888,891,981
biography, 415
Gay-Lussac, 528
Gegenbauerfunctions,419
Gegenbauerpolynomials, 185
generallineargroup,655
INDEX
1013
representation, 856-859
generalrelativity,918-932
generalized
eigenvector,
468
generalized
Fourier
coefficients, 150
generalizedfunctions,165-169
generalizedGreen's identity, 561, 574,
597
generating
function,
189
geodesic,905-908
relative
acceleration,
913
geodesicdeviation,913-918
equationof, 913
geodesic
equation
massiveparticles,926, 927
masslessparticles,926,928
geometricmultiplicity, 115
Gibbs,463, 807
Gibbsphenomenon,205-207
GL(n,
lll.) as aLie group,816
GL(V)
as aLie group,816
representation of, 856
GOde1,799
Gordan,996
biography,704
gradient
for Hilbertspaces,976
gradient
operator,
901
Gram, 28
Gram-Schmidt process,27-29
Gram-Schmidtprocess, 172
graph,5
Grassmann,743, 996
Grassmannproduct,740
gravitatioualred-shift,930
gravity
andcurvature,914
Green'sfunction,289
adjoint,566
advanced,634
and
Fourier
transform,
628
construction,
569
diffusion
operator,
632
DirichletBC
circle,618
DirichletBVP,613
eigenfunctionexpansion, 577-579
1014
INDEX
for SOLDO,
565-577
for the Laplacian,
595-596
formal
considerations,
557
Helmholtz operator, 630
in
20,
643
inonedimension, 553
indefinite, 554
inhomogeneous
Bes,
574
iutegral equations, 600
Laplacian, 629
NeumannBVP,620
exterior, 621
interior,
620
physical interpretation, 577
properties, 567
retarded, 634
second order DO, 562
synnnetry, 567
wave
equation,
633
Green's
function
for
dhlx,
554
for d
2
/dx
2,
555
Green's identity, 567, 596, 623, 627
Green, biography, 561
group,
652-656
abelian, 654
affine, 876
algebra, 687
symmetric gronp, 718
automorphism, 655
center of a, 658
commutative, 654
commutator, 657
homomorphism, 655
isomorphism, 655
left action, 663
multiplicatiou, 652
multiplication table, 656
of affine motioos, 817
order of, 652
orthogonal, 656
realization, 664
representation
definition, 674
irreducible,
projection
operator,
698
matrix,675
particles and fields, 699
teusorproduct, 699
right action, 663
rigidrotations, 657
special orthogonal, 657
special unitary, 657
symplectic, 657, 749
unitary, 656
group action, 663-664,
817-820
effective, 663
infinitesimal
generator,
827
transitive, 663, 818
guidedwaves
TE,533
TEM,533
TM,533
Haar measure, 832
Halley, 422
Harrrilton,
178,490,996
biography, 806
Hamiltonian system, 805
Harrriltonian vector field, 805
Hankel function, 425
asymptotic
expansion of, 316
Hankel transform, 434
harmonic functions, 236
heat equation, 327, 591, 621
synnnetry group, 957-961
heat transfer, 528
steady-state, 528
time-dependent, 530
heat-conducting plate, 528
Hegel,738
Heisenberg, 64, 167
helicity, 876
Helmholtz, 178, 588, 849
Helmholtz equation, 542
Helmholtz
operator
Green's
function
for,630
Hermite, 183,798
biography, 63
Hennitepolynomials, 176,
180, 517
hermitian
conjugate,
61
hermitian kernel, 497-501
hermitian operator,
63-67
llilbert,
11,28,
199,463,704,799,848,
920,996
biography,
lSI
Hilbert space,145-157
basis, ISO
definition, 148
differential of functions, 976
functions
on
derivative
of, 974
Hilbert transforms, 308
Hilbert-Schmidt operator, 465
Hilbert-Schmidt theorem, 498
Hilbert-Schmidtkernel, 466, 494
Hilbert-Schmidt operator, 847
Hodge star operator, 756-758, 795
HOlder,463
homogeneous SOLDE
exact,364
second
solution,
358
homographic transformations, 239
homomorphism
algebra, 43
kernel,658
Lie group, 816
synnnetric, 655
trivial, 655
hyperbolic PDE, 589, 626-628
hypergeometric DE, 407
hypergeometric function,
413-419
contiguous, 417
Eulerformula, 436
integral representatioo of, 434
hypergeometric
series,414
hypersurface, 583
ideal,44
idempoteot, 688
esseotially, 689
primitive, 689
identity
additive, 20
multiplicative, 20
ideotity
map,S
identity
operator,
50
identity
representation.
674
iguorable coordinate, 593
imageof a
subset,
5
implicit function theorem, 350
iodicial polynomial, 406
INDEX
1015
induced
representations,
871
induction
principle,
12
inductive definition, 14
inequality
Bessel,I49
Darboux, 245
Parseval,I49
Schwarz, 29
triangle, 31
infinitesimal actioo, 826-831
adjoint, 827
infinitesimal generator, 827, 830
inhomogeneous B'Cs,
574-577
inhomogeneous SOLDE
general solution, 360
initial value problem, 559, 583
numerical solution, 383
injectivemap,6
inner automorphism, 825
inner
product,
23-32
braandket
notation,
25
definition of, 24
norm and, 31
positive definite, 25
pseudo-Riemannian.
25
Riemannian,
25
sesqnilinear, 25
inner
product
space,
25
integrability condition, 888
integral curve, 783
integral
equation
characteristic value, 489
Fredhohu,494-504
Green's
functions,
600
kernel of, 488
of
the first kind, 488
of
the second kind, 488
Volterra, 488
Volterra,
of secondkind
solution, 490
integral operator, 452
integral transforms, 433
Bessel function, 434
integration
onmanifolds, 800-801
integration
operator,
73
interior
product,
793
intersection,
2
1016
INDEX
intrinsic
spin,999
invariant,
937
invariant map, 937
invariant
operator
matrix
representation,
112
invariant subspace, 112-114,676, 677
definition, 112
inverseimage, 5
inverse
mapping
theorem,
777
inverse
of a
map,
6
inverse
of
a matrix, 97-100
inversion, 238, 912
irredncibility criterion, 685
irreducible
representation,
677
i-throw
functions, 696
norm
of
functions, 696
irreducible
tensor
operators,
705-707
isometric
map,
908
isometry, 90&-912
time
translation,
923
isomorphism,
36
algebra, 43
Lie algebra, 822
Lie group, 816
Jacobi,
183,416,490,615,
663, 702, 704,
738,807,888
biography, 177
Jacobi functions, 418
Jacobi identity, 782, 790, 826
Jacobi polynontials, 176,
184,418
special
cases,
176
Jacobian matrix, 777
Jordan arc, 242
Jordan canonical form, 485
Jordan's lemma, 275
Kan~
738
Kelvin, 561
Kelvin equation, 538
Kepler problem, 1001
kemel, 35
degenerate, 501
integral operator,
452
integral transforms, 433
separable, 501
Killing, 743, 942
biography, 842
Killing equation, 909
Killing form, 841, 845
Killing form
of
glen,
R), 844
Killing vector field,
90&-912, 923, 926,
930
conformal, 911
Kirchhoff, 588
Klein, 388, 743,
798,849,891,920,942,
996
Klein-Gordon equation, 328
Korteweg-de Vries equation, 971
Kovalevskaya, 463
biography, 587
Krarners-Kronig relation,
309
Kronecker,
11,96,388
hiography, 738
Kronecker delta, 26 .
Kummer, 30, 704, 738, 842, 891.
KUlla,biography, 390
Lagrange, 96, 178, 182, 198,
199,415,
423, 528, 704
biography, 984
Lagrange identity, 366, 434, 525, 560
Lagrange multiplier, 991
Lagrange theorem, 670
Lagrangian, 804, 980
null, 987
Laguerrepolynontials, 176,
180-181
Laplace, 28,
199,614,806,920,982,985
biography, 527
Laplace llansfonn, 433
Laplace's equation, 327
Cartesian
coordinates,
526
cylindrical coordinates, 535
spherical
coordinates,
541
Laplacian
Green's function for, 595, 629
separation
of
angularpart, 331
Laurent series, 252-262, 606
construction,
254
uniqueness, 258
Laurent, biography, 271
Lavoisier, 95, 985
left coset, 658
left ideal, 44, 689
minimal, 689
left translation, 820
as action, 827
left-invariant
l-form, 820
left-invariant
vectors,
820
Legendre, 177,
199,490,614,888
biography, 182
Legendre eqnation, 517
Legendre functions, 419
Legendre polynontials, 155, 182-184,
340,359
and delta function, 188
and Laplacian, 188
Legendre
transformation,
804
Leibniz, 95, 738
Leibniz rule, 15
length of a vector,
30-32
Levi-Civita, 891
biography, 898
Levi-Civita tensor, 745, 869
Lie,711,743,798,842
biography, 941
Lie algebra,
833-845
abelian, 834
adjoint map, 825
Cartan
metric
tensor,
843
Cartan theorem, 844
center, 834
commutative,
834
compact, 843
decomposition,
844
derivation, 840
ideal, 834
Killing form of a, 841
of a Lie gronp,
820-826
of nnitary gronp, 824
of vector fields, 782
representation, 859--876
definition, 845
semisimple, 844
simple, 844
structure
constants,
834
Lie
algebra
of SL(V), 823
Lie bracket, 782
Lie derivative, 788
INDEX
1017
covariant
derivative,
902
of a I-form, 790
of vectors, 789
Lie
derivative
of
p-forms, 793
Lie gronp, 815
compact
characters, 853
matrixrepresentation, 852
representation, 852
unitary
representation,
846
Weyloperator, 847
homomorphism, 816
integration,
832-833
density function, 833
local,817
representation,
845
Lie mnltiplication, 833
Lie snbalgebra, 834
Lie's
first
theorem,
829
Lie's second
theorem,
826
Lie's third theorem, 826
ligbtcone, 837
linear
combination,
21
linearfunctional, 34, 455
linearindependence, 21
linear
operator,
33
nnll space of a, 35
linearPOE, 584
linear transformation,
32-40
bonnded,454
definition,
33
pullback of a, 39
Liouville, 512, 654
biography, 514
Liouville
substitution,
513,520
Liouville's
theorem,
807
Lipschitz condition, 351
little algebra, 871
little group, 871, 873
local diffeomorphism, 768
local group of transformations, 817
local Lie group, 817
local operator, 452
logarithntic function, 295
Lorentz, 799
Lorentz algebra, 864
1018
INDEX
Lorentz
forcelaw,794
Lorentz group, 657, 837
Lorentz metric, 892
Lorentz
transformations,
837
loweringindices, 750
lowering
operator,
335
Maclaurin
series,
254
manifold, 764
coordinate functions, 764
ffat, 895
orientable, 800
product, 767
pseudo-EUemanmrran,887
symplectic, 801
map,
4-7
bijective,6
codomain,S
dlfferentiable,768
domain, 5
equality of, 5
functions and, 5
graph
of
a, 5
identity, 5
imageof a
subset,
5
injective,
6
inverse
of a,6
one-to-one,6
onto, 6
range
of
a, 5
surjective, 6
target space, 5
matbematical induction, 12
matrix,
82-86
antisymmetric, 87
basis
transformation,
91
complex
conjugate
of, 87
diagooal, 88
hermitian, 87
hermitian
conjugate of, 87
inverseof a,
98
irreducible, 112
operations
ona,87
orthogonal, 87
rank
of
a, 86
reducible, 112, 113
representation
orthonormal basis, 89
row-echelon,99
symmetric,
87
transpose
of a, 87
triangular, 99
unitary, 87
matrix algebra, 42
maximally
symmetric spaces, 910
Maxwell's equations, 796
Melliu transform, 434
Mendelssobn,615,738
meromorphic functions,
293-294
metbod
of
images, 616
sphere, 617
methodof steepestdescent,521
metric
space,7-10
convergence, 9
definition, 8
minimal ideal, 44, 856
Minkowski, 920
Minkowski metric, 892
Minkowski space, 754
Mittag-Leffler, 388, 463, 588
Mittag-Leffler
expansion,
294
modified Bessel functions, 425
moment
of
inertia
matrix,
88
Monge, 95, 198
Morera's
theorem,
252
multilinear mapping,
729-735
tensor-valued, 735
multiplicative identity, 20
multivalued functions,
295-302
n-equivalent functions, 945
a-sphere, 764
n-tbjet
space, 945
Napoleon, 198, 527
natural isomorphism, 733, 756
natural pairing, 731
neighborhood, 459
Neumann, 177, 702
biography, 620
Neumann
Be,
591
Neumann
Bvp,
591, 619-621
Neumann function,
424
Nemnarm series, 493, 601, 602
Newton,
330,415,
527,798,
806,982
Newtoniangravity,916-918
Noether,704
biography,996
Noether's
theorem,
995
non-localpotential,631
norm
of a
vector,
30
operator,454
product
of
operators,
456
normaloperator,117
nornuU
snbgronp,659
normal
vectors,
26
normed
linear
space,31
normed
vectorspace
compact
subsetof, 461
noll divergence,993
noll Lagrangian,987
noll space,35
noll vector,751, 837
numerical
integration,
74-76
Simpson's 1/3rule, 75
Simpson's 3/8rule, 75
trapezoidalrule, 75
Ohm,614
Olbers,422
one-form, 786
one-parameter
group,
784
openball, 459
open subset,459
operations
on
matrices,
87-89
operator,
33
adjoint,61
existenceof, 457
angularmomentrun, 331
annihilation,374
bouuded,454
closed,508
bounded,508
compact
Hermitian
spectraltheorem,476
compact
normal
spectraltheorem,477
compact
resolvent,
508
creation,374
diagonalizable,116
differentiation,
73
INDEX
1019
domainof, 507
eqnality,49
evolution, 58
expectation
valueof, 64
extension of, 508
finiterank,464
formally self-adjoint,597
functions of an, 53
hermitian,63, 508
eigenvalue,
117
hermitian
conjugate
of, 61
Hilbert-Schmidt, 465, 511
integration,
73
inverse
of an,50
kernel of an, 35
local,452
negative
powers
of,52
norm of an, 454
normal,117
diagona1izable, 119
eigenspace of, 118
noll spaceof an,35
numerical
analysis,
70
polarizationideotity,50
positive,65
projection,
67
projection
orthogonal,68
pollbackof an,39
raising
and
lowering,
335
regularpoint,457
representation
of an,83
right-shift,453
scalar,706
self-adjoint,508
spectrnm,457,457-458
spectrnmof an, 115
square
rootof, 127
Stnrm-Liouville,510
trace of an, 102
unbounded,507
mlitary, 66, 127
eigenvalue,
117
operatoralgebra,49-76
operatorpolynomials,51-53
operator
spectrum,
457
o(p, n -
pl,
837
1020
INDEX
optical theorem, 308
orbh,
663,676,818
orbital angular momentum, 999
ordered pairs, 2
orieiltable manifolds, 800
orientation,
745,746
positive, 746
orientedbasis,745
orthochronous,
837
orthogonal groop, 656, 824
Lie algebra of, 824
orthogonal polynomials, 153
classical, 173
classification, 176
differential equation, 174
generating functions, 189
recurrence
relations, 176
expansion, 186
least sqnare fit, 155
orthogonal
transformation,
97
orthogonal vectors, 26
orthogonality,
26-27
orthonormal basis, 26
orthonormal
frames,
887
p-forms, 741
pairing
natural, 731
parabolic PDE, 589,
621-{j26
parallel transport, 906
parallelogram law, 31
parity,667
Hermitepolynomials, 194
Legendrepolynomials, 194
Parseval inequality, 149, 851
Parseval's relation, 223
partial DE,
584-592
characteristic
systemof, 939
order of, 584
principal part, 584
particle in a box, 531
particle in a cylindrical can, 539
particle in a bard spbere, 543
partition,4,669
past light cone, 838
Pauli spin matrices, 88, 835, 840
Peano, 799
periodic
Be,515
permutation
even, 668
odd,668
parity of, 667
permutation group, 664
permntation symbol, 93
permutation
tensor,
754
perturbation theory, 603-{jlO, 697
degenerate, 609
first-order, 608
nondegenerate, 607
second-order, 608
Peter-Weyl theorem, 853
Peter-Weyl
theorem
Fourier
series,853
photon
capture
cross
section,
929
Planck,388,463,920
Poincare,63,478,497,620,744,920
biograpby, 797
Poincare algebra, 840,
844
representation, 868-876
Poincare group, 657, 817, 839, 872
Poincare
lemma,
797
converseof, 797
Poisson,
178,512,528,614,653
Poisson bracket, 808
Poisson'sequation, 327, 917
polar decomposition, 129-130
pole, 273
polynomials
orthogonal, 153
positive
operator,
65
positiveorientation, 746
potential
non-local, 631
separable, 631
power
series,
252
differentiation of, 253
integration of, 253
uniform
convergence,
253
p(p,
n -
p),
837
preimage,5
primitive idempotent, 689
principal value, 285
productmanifold, 767
projectable symmetry, 944
projection operator,
67-70, 474, 604
completeness
relation,
68
orthogonal, 68
projective
group
density function, 833
one-dimensional, 820
projective space,4
prolongation, 945-950
of
a function, 946
of
groups, 948
of
vector fields, 949
propagator, 626
Feynman, 636
proper
orthochronous, 837
proper
subset,
2
Priifer substitution, 518
Puiseux, biography, 296
pullback,735,786,791,801
linear
transformation,
39
pullback map, 791
pullback
of
p-forms, 741
quantization
harmonic
oscillator
algebraic, 375
analytic, 374
hydrogen atom, 422
quarks,70I,703,873
quaternions,
44
quotient group, 660
quotient
set,4
raising indices, 750
raising operator, 335
range
of
a map, 5
rank
of
a matrix, 86
rational
function,
274
and trig functions
integrals of, 279
integration of, 276
rational
numbers
densesubsetof reals,460
real
coordinate
space,
21
real vector space, 20, 130-137
realization, 664
reciprocal lattice
vectors, 208
recurrence
relations,
153
INDEX
1021
redshift,
930
reducedmatrixelements,707
reducible
representation,
677
regular graphs, 713
regularpoint, 233
regular
representation,
686
regular singularpoint
second-order linearDE, 405
relation, 3
relativeacceleration, 913
relativisticelectromagnetism, 792
representation
abelian groop, 681
adjoint, 679, 704
carrier
space,
674
cbaracter, 683
compact Lie group, 852
complex conjugate, 679
dimensionof,
674
direct sum, 679
equivalent, 674
faithful, 674
geuerallinear
group,
856-859
group, 674
identity, 674
irreducible, 677
Kronecker
product,
700
Lie
group, 845
operators, 83
reducible, 677
regular, 686
symmetric group,
707-723
tensor prodoct, 700
antisynunetrized, 701
character, 700
synunetrized, 701
trivial, 679
unitary, 678
compact Lie group, 846
representation of
g[(n,lR), 860
s[(n, It), 861
so(3),864
so(3,
I),
867
su(n),861
u(n),861
residue, 270-273
1022
INDEX
definite integrals, 275
definition, 271
residue
theorem,
271
resolution of identity, 482, 689, 721
resolvent,
481J--485
compact, 508
Laurent
expansion,
481
resolventset.457
openness of, 460
resonant
cavities,533
Riccati equation, 968
Ricci, 891, 898
Ricci tensor, 917, 919, 920
Riemann, 30, 199,296, 704,
798,849,
891,981
biography, 888
Riemann
curvature
tensor,
889, 904
Riemann normal coordinates, 915
Riemann surfaces, 296-302
Riemannian manifold,
887-896
right coset, 658
right ideal, 44
right translation, 820
right-invariant l-form, 820
right-invariant vectors, 820
right-shift operator, 453
eigenvalues
of,458
rigid rotations, 657
Rodriguez formula, 174
Rosetta stone, 198
rotation algebra, 864
, rotation gronp, 675, 862
character, 865
rotation
matrix,
865
Runge, biography, 388
Runge-Kntta method,
387~391
Rnssell, 11, 799
saddle point approximation, 313
scalar operator, 706, 707
scale
transformations,
819
Schelliog,738
Schmidt, biography, 28
Schopeohaner,738
Schriidinger, 64, 807
SchrOdinger equation, 675
Schriidinger equation, 58, 328, 410, 420,
531,543,631
classical limit, 383
one
dimensional,
381
Schur, 711, 850, 874
biography, 681
Schur's lemma, 680, 681,
707,845,861
Schwarz, 463, 739
biography, 30
Schwarz inequality, 29, 149, 152
Schwarzretlection principle, 305-306
Schwarzschild geodesic, 925-932
Schwarzschildmetric, 892
Schwarzschild
radius,
925
Schwarzschild, biography, 919
second
order
linear
DE
Sttum-Liouvi11esystems, 513
second order POE, 589--592
hyperbolic, 589
second-orderPDE
classification, 589
elliptic, 589
parabolic, 589
ultrahyperbolic, 589
selectionrules, 701
self-adjoint
operator,
63
self-adjoint SOLDOs,
564-565
semisimple Lie
algebra,
844
separable kernel, 501
separable potential, 631
separated
boundary
conditions,510
separation
of
variables,
328
Cartesian,
526-534
cylindrical,
535-540
spherical, 540-544
separation theorems,
361-363
sequence,
9
Canchy,1O
complete
orthonormal,
149
series
Fourier-Bessel, 536
~eornann,601,602
sesquilinear
inner
product,
25
set,
1-3
Cantor, 12
complement
of, 2
countably infinite, 12
element
of, 1
empty, 2