Hassani S. Mathematical Methods: For Students of Physics and Related Fields
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824 INDEX
heat-conducting rod, 662
Heaviside, 370
Heaviside, Oliver
biography, 382
Heisenberg, 26, 151, 675
Heisenberg uncertainty relation, 699
Helmholtz Coil, 291–293
Helmholtz free energy, 54
Hermite DE
recursion relation, 668
Hermite polynomial, 670
orthogonality, 672
Hermite polynomials, 229, 575
generating function, 673
Hermite, Charles
biography, 674
HNOLDE, 575
characteristic polynomial, 576
homogeneous
function, 57
homogeneous function, 57–59
homogeneous SOLDE
exact, 571
Hooke, 97
Hopf bifurcation, 770
HSOLDE, 564
second solution, 568
Huygens, 97, 103
hydrogen atom, 677–680, 802
hyperbolic cosine, 290
hyperbolic sine, 290
hypergeometric function, 328–330
confluent, 332–333
integral representation, 329
identity matrix, 180
indeterminate form, 294–297
index
free, 440
indicial equation, 609
induction
mathematical, 265–266
inductive definition, 266
infinite series, 266–274
inner product, 218–222
positive definite, 187
Riemannian, 187
inner product matrix, 185
integral, 79
as function, 317–326
Bessel’s, 652
derivative of, 85–86
function of trigonometric, 534–536
indefinite, 87
line, 387–391
Mellin inversion, 722
rational function, 529–531
rational trigonometric, 532–534
integral transform, 693
kernel, 693
integrand, 80
integrating factor, 553–555
integration, 77–80
application
Cartesian coordinates, 104–107,
112, 115–117
cylindrical coordinates, 107–109,
112–115, 118–119
double integrals, 115–122
electricity, 104–109
general, 91–96
gravity, 104–109
magnetostatics, 109–115
mechanics, 101–103
single integral, 101–115
spherical coordinates, 120–122
triple integrals, 122–128
Cauchy integral formula,
508–509
change of dummy variable, 82
complex function, 503–508
interchange of limits, 82
linearity, 82
parameter, 80
partition of range, 82
point, 79
properties, 81–89
region of, 79
small region, 83
symmetric range, 84
transformation of variable, 83
variable, 80
intersection, 783
inverse
matrix, 203, 207
of a matrix, 180
inverse Fourier transform, 695
ionic crystal
one-dimensional, 145
potential energy, 164
two-dimensional, 157
INDEX 825
ISOLDE, 569
isoperimetric problem, 738
iterated map, 754–763
fixed point, 755
orbit, 755
Jacobi, 211, 331
Jacobi, Carl Gustav Jacob
biography, 326
Jacobian, 207–210
in probability, 804
Jacobian matrix, 208
Kaluza, 215
Kepler, 89, 97
Kepler’s first law, 582
Kepler’s second law, 582
Kepler’s third law, 583
Kepler, Johannes
biography, 579
kernel
integral transform, 693
Klein-Gordon equation, 747
Koch snowflake, 777
Kronecker delta, 222, 442, 449, 489
Euclidean metric, 466
generalized, 452
Lagrange, 294, 304, 326, 330, 594,
617, 642
biography, 742
Lagrange identity, 572
Lagrange multiplier, 360, 738
Lagrangian, 740–745
interacting particles, 741
Klein-Gordon, 747
particle in EM field, 746
single particle, 741
Lagrangian density, 744–745
electrodynamics, 745
Laguerre polynomials, 230, 679
Laplace, 115, 304, 322, 326, 744
Laplace transform, 712–723
and differential equations, 718–721
Bromwich contour, 722
convolution, 716
cosine, 713
derivative, 717–718
first shift, 714
gamma function, 713
imaginary exponential, 713
integral, 717–718
inverse, 721–723
linearity, 714
Mellin inversion integral, 722
periodic functions, 716
properties, 713–717
second shift, 714
sine, 713
step function, 713
unit function, 713
Laplace’s equation, 411, 542, 546
Bessel functions, 642–654
Cartesian coordinates, 594–603
cylindrical coordinates, 639–656
Legendre polynomials, 610–617
radial equation, 619–622
solution
uniqueness, 592
spherical coordinates, 607–634
uniqueness of solution, 592–593
Laplace, Pierre Simon de
biography, 593
Laplacian, 411
and Dirac delta function, 412
curvilinear coordinates, 435
Green’s function, 708–709
Laurent series
complex, 518–522
Lavoisier, 744
law of addition of velocities, 237
law of large numbers, 809
law of motion
relativistic, 253–254
Legendre, 304, 326
Legendre equation, 575
recursion relation, 611
Legendre functions
second kind, 618
Legendre polynomial, 228, 614, 616
expansion in, 628–630
physical examples, 631–634
generating function, 621
Laplace’s equation, 610–617
multipole expansion, 621
orthogonality, 625
parity, 622
properties, 622–628
recurrence relation, 623
Rodrigues formula, 626
826 INDEX
Legendre polynomials, 229, 575
Legendre transformation, 54, 748
Legendre, Adrien-Marie
biography, 617
Leibniz, 46, 87, 90, 97, 210, 272, 482
Leibniz, Gottfried Wilhelm
biography, 103
length element
primary, 59
Levi-Civita symbol, 453
Levi-Civita symbols
cross product, 458
l’Hˆopital’s rule, 294–297
limit cycle, 770
line integral, 387–391
linear combination, 173
linear dependence, 174
linear equation, 230–234
compatible, 231
echelon form, 232
homogeneous, 234
incompatible, 231
linear independence, 174
linear operator, 216
linear transformation, 216–218
Liouville substitution, 588
logistic map, 755
second iterate, 757
Lorentz gauge, 419
Lorentz transformation, 243–247
general, 244
in 2 dimensions, 245
lowering indices, 457–459
Lyapunov exponent, 763
Maclaurin, 210, 272
Maclaurin series, 287
Madelung constant, 165
magnetic charge, 409
magnetic dipole moment, 410
magnetic field
moving charge, 30
magnetic force
current loop, 420
moving charge, 30
magnetic monopole, 409
manifold, 456, 469
map
iterated, 754–763
marginal probability, 786–789
mathematical induction, 265–266
matrix, 177
G-orthogonal, 191, 222
space, 200
identity, 180
inner product, 185
inverse, 180, 203, 207
Jacobian, 208
metric, 185
multiplication rule, 442
orthogonal, 190
symmetric, 182
transformation
in space, 195
transpose, 181
unit, 180
zero, 180
Maxwell, 26, 369, 382
Maxwell’s equations, 415–419
derivation of wave equation, 417
relation to relativity, 237
Maxwell, James Clerk
biography, 419
Maxwell-Boltzmann statistics, 791
mean, 790
Mellin inversion integral, 722
membrane, 686–687
metric connection, 465–468
relativity, 466
metric matrix, 185
metric tensor, 454–461
definition, 456
relativity, 458
minimal coupling, 749
Minkowski, 215
mode
of oscillation, 682
M¨obius band, 366
moment
quadrupole, 449
moment generating function, 790
binomial distribution, 793
Poisson distribution, 799
moment of inertia, 122
momentum
generalized, 748
relativistic, 249
zero mass particle, 250
Monge, 115, 304
INDEX 827
motion
constant of, 552
multipole expansion, 297–299
Napoleon, 304, 593
natural frequency, 586
Neumann function, 645
Newton, 16, 26, 43, 46, 78, 87, 90, 103,
122, 272, 294, 317, 322, 326,
330, 481, 482, 548, 593
Newton, Isaac
biography, 96
NOLDE, 575
nonautonomous, 767
normal distribution, 806–809
sum of two, 807
nth iterate, 760
ODE, 542
ODE and PDEs, 542–550
Olbers, 641
operator
angular momentum, 412
spherical coordinates, 435
del, 359
differential, 576
linear, 217
linear, 216
orientable surface, 366
orthogonal
matrix, 190
orthogonal polynomial
standardization, 227
orthogonal polynomials, 227–230
orthonormal
basis, 186
parabolic coordinates, 73
paraboloidal coordinates, 74, 437
parallel translation, 465
Parseval relation, 654
Parseval’s relation, 724
partial derivative, 47–59
particle in a box, 675
Pascal, 15, 103, 481
passive transformation, 178
PDE, 542
separation
Cartesian coordinates, 544–546
cylindrical coordinates, 547–548
spherical coordinates, 548–550
PDE and ODE, 542–550
period-doubling, 757
periodic BC, 574
permutation, 791
phase space, 764–766
diagram, 764
trajectory, 764
Planck, 666
plane
basis, 175
Poincar´e, 674
Poisson, 594
Poisson distribution, 797–800
Poisson equation, 411, 542
astrophysics, 415
Green’s function, 709
polar coordinates, 16
polar equation, 549
pole
of order m, 528
simple, 528
polynomial
Hermite, 229, 670
Laguerre, 230, 679
Legendre, 228, 229, 614, 616
Laplace’s equation, 610–617
orthogonal, 227–230
standardization, 227
position vector, 19
potential, 21–28, 399
centrifugal, 581
difference, 399
of a dipole, 299
potential energy, 553
power series, 283–299
continuity, 285
differential equations, 307
differentiation, 285
integration, 285
operations, 520
radius of convergence, 283
zero, 285
pressure, 46
primary curve, 59
primary surface, 60
probability
average, 790
basic concepts, 781–792
binomial distribution, 792–797
828 INDEX
conditional , 786–789
correlation, 803
covariance, 803
density, 801
expectation value, 790
independent random variable, 802
marginal , 786–789
mean, 790
moment generating function, 790
Poisson distribution, 797–800
sample space, 784–786
set theory, 782–784
standard deviation, 790
variance, 790
probability space, 784
prolate spheroidal coordinates, 74, 213,
437
proper time, 239–240
quadrupole moment, 449
quantization
hydrogen atom, 679
quantum harmonic oscillator, 667–674
quantum mechanics
angular momentum operator, 412
spherical coordinates, 435
quantum particle
in a box, 675–677
quantum tunneling, 676
quaternions, 11
radial, 19
radial equation, 549
raising indices, 457–459
random event, 781
random variable
continuous, 801–809
independent, 802
transformation, 804–806
rate of change, 44
ratio test
Waring, 273
recursion relation, 308, 610
relativistic collision, 250–253
relativistic energy, 249
relativistic law of motion, 253–254
relativistic momentum, 249
relativity
geodesic, 466
metric connection, 466
metric tensor, 458
principle, 238
special, 237
residue, 526
calculus, 525–536
residue theorem, 527
definite integral
rational function, 529
rational trigonometric, 532
trigonometric function, 534
retarded Green’s function, 712
Ricci tensor, 470
Riemann, 321
Riemann curvature tensor, 468–471
Riemann zeta function, 269
Riemannian manifold, 456
right-hand rule, 392
rigid transformation, 190
Rodrigues formula, 626
Rosetta stone, 304
row vector, 181
sample space, 784–786
Savart, Felix
biography, 115
scalar curvature, 470
scalar function, 445
Schr¨odinger, 675
biography, 666
Schr¨odinger equation, 543, 546, 666–680
time-independent, 666
Schwarz inequality, 185, 220
Schwinger, 26
second iterate, 757
second variation, 735–738
self-similarity, 775
separated boundary conditions, 574
separation of time, 543
separatrix, 766
sequence, 259–262
bounded, 261
convergence, 260
Cauchy criterion, 261
divergence, 260
functions, 274–279
limit, 260
monotone decreasing, 261
monotone increasing, 261
partial sum, 259, 267
series, 266–274
INDEX 829
alternating
test, 270
application to DE, 307–311
complex, 518
Laurent, 518–522
Taylor, 518–522
convergence
absolute, 268
comparison test, 268
conditional, 272
generalized ratio test, 270
integral test, 268
n-th term test, 267
ratio test, 269
convergent
grouping, 273
rearranging, 273
familiar functions, 287–291
Fourier, 299–303
complex numbers, 489
Fourier–Bessel, 655
functions, 274–279
uniform convergence, 276
geometric, 271
harmonic
order p, 269
Laurent
complex, 518
Maclaurin, 287
binomial function, 288
complex, 518
exponential function, 287
hyperbolic function, 289
logarithmic function, 291
trigonometric function, 287
operations on, 273–274
power, 283–299
differential equations, 307
Taylor, 286–287
complex, 518
multivariable, 305–307
uniform convergence
differentiation, 278
integration, 278
uniformly convergent, 277–279
set theory, 782–784
complement, 783
difference, 783
disjoint sets, 783
intersection, 783
union, 782
Venn diagrams, 783
sine transform, 697
soap film problem, 733
SOLDE, 563–570
basis of solutions, 565
central force, 579
constant coefficient, 575–587
homogeneous, 576–583
inhomogeneous, 583–587
homogeneous, 564
second solution, 567–569
inhomogeneous
general solution, 569–570
Kepler problem, 580
linearity, 564–565
normal form, 563
singular point, 563
superposition, 564–565
superposition principle, 564
uniqueness of solution, 564–565
uniqueness theorem, 565
variation of constants, 569
Wronskian, 566–567
solid angle, 344–350
total, 349
source point, 25, 79
space
dimension, 11
flat, 470
point, 11
probability, 784
spacetime distance, 240–242
being zero, 242
span, 175
special functions, 550
standard basis, 216
standard deviation, 790
statistical independence, 788
statistics
Bose-Einstein, 792
Fermi-Dirac, 791
Maxwell-Boltzmann, 791
stellar equilibrium, 415
step function, 152–153
Dirac delta function, 153
Laplace transform, 713
Stifel, 481
Stirling, 320
Stirling approximation, 319, 792, 808
830 INDEX
Stokes’ theorem, 391–398
Stokes, George Gabriel
biography, 398
strange attractor, 778
Sturm–Liouville
system, 574
Sturm-Liouville equation, 574
subset, 782
success excess, 793
summation, 262–266
superposition principle, 25, 564
surface
primary, 60
Sylvester, and Cayley, 192
Sylvester, James Joseph
biography, 210
symmetric matrix, 182
symmetric tensor, 452
Taylor series, 286–287
complex, 518–522
multivariable, 305–307
Taylor, Brook
biography, 294
tensor, 447–454
addition, 450
algebraic properties, 450–452
contraction, 451
differentiation, 462–468
Einstein curvature, 471
electrodynamics, 459–461
energy momentum, 471
Levi-Civita symbols, 453
metric, 454–461
definition, 456
relativity, 458
multiplication, 451
numerical, 452–454
rank of, 448
Ricci, 470
Riemann curvature, 468–471
symmetrization, 452
torsion, 463
terminal velocity, 559
theorem
central limit, 809
time
coordinate, 239–240
proper, 239–240
time constant, 586
toroidal coordinates, 74, 213, 437
torque, 28
torsion tensor, 463
transfer function, 586
transform
cosine, 697
Fourier, 693–712
and quark model, 702
application to DEs, 702–704
convolution theorem, 724
examples, 698–702
Gaussian, 699
Green’s functions, 705–712
heat equation in 1D, 704
inverse, 695
of delta function, 698
properties, 696
integral, 693
Laplace, 712–723
and differential equations, 718–
721
Bromwich contour, 722
convolution, 716
cosine, 713
derivative, 717–718
first shift, 714
gamma function, 713
imaginary exponential, 713
integral, 717–718
inverse, 721–723
linearity, 714
Mellin inversion integral, 722
periodic functions, 716
properties, 713–717
second shift, 714
sine, 713
step function, 713
unit function, 713
sine, 697
transformation
active, 178
coordinate, 13
differentiation, 197
gauge, 418
Legendre, 54, 748
linear, 216
Lorentz, 243–247
matrix
in space, 195
orthogonal, 442
INDEX 831
passive, 178
rigid, 190
transient term, 586
transpose
of a matrix, 181
transposition, 181
properties, 182
triangle inequality, 480
tunneling, 676
uncertainty relation, 699
uniform convergence
Weierstrass M-test, 276
uniformly convergent series, 277–279
union, 782
unit matrix, 180
unit vectors, 5
universal set, 782
partition, 785
Van de Graff, 117
Vandermonde, 210
variable
random
continuous, 801–809
transformation, 804–806
variance, 790, 801
variational problem, 728–740
constraints, 738–740
several dependent variables, 734
several independent variables, 734
soap film, 733
vector
Cartesian
component, 216
n-dimensional, 216
column, 177
component, 176
contravariant, 445–447
coordinate system, 16–31
covariant, 445–447
cross product, 7–10
field
conservative, 398–404
curl, 391–398
flux, 365–369
G-orthogonal, 219
in space, 199
indices, 439–471
inner product, 182–191, 198–202
plane, 3–10, 174–191
position, 19
row, 181
space, 3–10, 192–207
time derivative, 350–355
transformation, 194–198
transformation of components, 176–
182
transformation properties, 441–445
unit, 5
vector analysis
curvilinear, 423–435
vector field
conservative
curl, 400
curl of, 394
divergence, 374
vector potential, 408
vector space, 173, 215–227
velocity, 44
terminal, 559
Venn diagrams, 783
vibrating membrane, 686–687
Vieta, 481
Wallis, 97, 293, 321, 326
Wallis, John
biography, 90
wave equation, 543, 680–687
advanced Green’s function, 712
from Maxwell’s equations, 417
Green’s function, 711–712
retarded Green’s function, 712
wave guide, 682–686
cylindrical, 686
longitudinal part, 682
rectangular, 685
transverse part, 682
weight function, 227
Wheatstone, 382
Wronskian, 566–567
Yukawa potential, 700
zero mass, 250
zero matrix, 180
zero spacetime distance, 242
zeta function, 269