Kusse B.R., Westwig E.A. Mathematical Physics: Applied Mathematics for Scientists and Engineers
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666
ERRATALIST
p.
75
Change Equation 4.45 to read:
(4.45)
p.
133
Exercise 25, first line, change
pc(z,
y,
y)
to
pc(2,
y,
z).
p.
152
In
all
equations
on
the page, change
gn
to
c,,
i.e.,
cc
f
(2)
=
C
~"(2
-
zo),
=
C,
+
C~(Z
-
2,)
+
C~(Z
-
2,)'
+
.
.
.
.
(6.96)
n=O
(6.97)
(6.98)
(6.99)
-
-
...+
c-2
+
c-1
(2
-
2,)2
+
co
+
c1(z
-
2,)
+
c2(z
-
zo)2
+ .
.
.
,
(2
-
2,)
p.
162
Paragraph below Equation 6.135,
(i) third line, change
C1
to
C2
and
C2
to
CI
.
(ii) seventh line, change
C2
to
C1.
p.
215
Exercise 49. (b), change
y
=
1/2 to
u
=
1/2.
p.
216
Exercise
55,
change the sentences beginning in the second line
of
the para-
graph to the end
of
the paragraph to read:
Integrate to obtain
2
=
&I).
p.
241
Change Equation 7.90
to
read:
p.
242
Change the first line
of
Equation 7.92 to read:
N-1
(7.90)
n=-N
ERRATA
LIST
667
p.
300
Exercise 21:
(i) Change the last sentence of the first paragraph to read:
“Show that the conditions for closure with zero contribution are different for a
real, definite integral than they are for a Fourier inversion.”
(ii) Change the first equation to read:
W2
w3
+i
F(w)
=
-
(iii) Change the equation under (b) to read:
p.
376
Figure 9.14, change the label on the abscissa from “imag” to “real”.
p.
378
Figure 9.16, change the label on the abscissa from “imag” to “real”
p.
354
Change the part
of
the sentence below Equation
10.104
to read:
“.
.
.where the
r,,
are unknown coefficents, and
s
is a fixed number, not necessarily
an integer or pure real. Often, however, it is a positive or negative integer.”
p.
379
Figure 10.16, change
-K
x(t)
to
read
-KO
x(t).
p.
383
Title
of
Figure 10.2
1,
change “Principal” to “Principle”.
p.
408
Exercise
22
(a), last line, change
IC
=
z,
to
3
=
yo.
p.
427
Exercise
53,
first line below the equation, change
a(z),
b(z)
top(z),
q(s).
p.
422
Exercise
54
(d), first line, change
d(t)
to
d(z).
p.
428
Change Equation 1 1.20 to read:
cc
III
,2
(11.20)
668
ERRATALIST
p.
448
Split
the
last sentence
of
the paragraph
Orthogonality
Relations
into two
sentences:
. . .
eigenfunction problem.
A
Hermitian operator can then be identified and the orthogonality properties
of
the
eigenfunctions determined.
p.
449
Change Equation 11.11 1
to
read:
p,
453
Change Equation 1 1.136 to read:
p.
462
Change Equation 1 1.176
to
read:
p.
463
(i)
Line above Equation 1 1.180, change
Be(
z)
to
Be
(
X) .
(ii) Line aboveEquation 11.181,change
Ao(z)
to
A,(%).
p.
464
Third paragraph, change “six” to “five”.
p.
473
Figure 11.21, last line
of
equation, change
+VO
to
-VO.
p.
474
Equation 1 1.237: replace closing brace by opening brace:
p.
482
Exercise 16(a), change Equation to read:
(1
1.11 1)
(1 1.136)
(1
1.176)
(1 1.237)
p.
484
Exercise 24, change the second and third line below the first equation to
read:
“. .
.
the differential equation
for
radially dependent component becomes”
ERRATALIST
669
p.
489
Exercise 34: insert
0
on the right-hand side of Laplace’s equation in two
dimensions:
p.
507
Exercise 7, change equation to read:
PX
q(z)
=
x
-
dt
(t
-
.)$(t).
J!
p.
508
Exercise 12.i., change equation to read:
1
?$(z)
=
x
dt
(.
-
t)”(t).
1,
p.
537
First line
of
the paragraph above Equation 13.45, change sentence to read:
“We should also
look
carefully at the little circular part.
. .
”
p.
542
(i)
Second line
from
top of page, change term
to
read
dz
=
e22.rrdr.
(ii) Change Equation 13.57 to read:
(13.57)
p.
547
Change Equation 13.72 to read:
(13.72)
dU
.i3v
aw
aw
ax
dz
ax
ag
-
+
2-
=
z
=
=
=
0
at the saddle points.
p.
581
Change Equation
14.100
to
read:
(14.100)
d
=
tZ-,
3
ax,
p.
593
Exercise
3,
change equation
to
read
p.
611
In the first line
of
the paragraph above Table 15.16, change “are shown”
to
“is
shown”.
p.
614
In the fourth line from the top
of
the second paragraph, change “dimension”
to “dimensions”.
670
ERRATALIST
p.
619
Table
15.25,
row
Cf],
column
94,
change
“-1”
to
“4’
p.
623
Change the sentence beginning in the 2nd line
of
the 2nd paragraph to read:
“The net result is that there
is
one
Dtl
representation,
no Dfl
representation, and
one
Drl
representation in the block diagonalization.”
p.
625
Equation
15.62,
change element
(1,3)
and
(3,l)
from
“1”
to
“-1”.
BIBLIOGRAPHY
Abramowitz, Milton, and Irene Stegun,
Handbook of Mathematical Function.
Dover Publica-
Arfken, George,
Mathematical Methods of Physics.
Academic
Press,
San Diego, CA, 1985.
Butkov, Eugene,
Mathematical Physics.
Addison-Wesley, Reading, MA, 1968.
Churchill, Rue1
V.,
and
James
Ward Brown,
Complex
Variables and Applications.
McGraw-
Griffel, D.H.,
Applied Functional Analysis.
John Wiley and Sons, New York, 1981.
Hamemesh, Morton,
Group Theory and its Applications to Physical Problems.
Adclison-
Hildebrand, F.B.,
Methods ofApplied Mathematics.
Prentice-Hall, Englewood Cliffs,
NJ,
1958.
Jackson, J.D,
Classical Electrodynamics.
John Wiley and Sons, New York, 1962.
Kaplan, Wilfred,
Advanced Calculus.
Addison-Wesley, Reading,
MA,
1973.
Mathews, Jon, and R.L. Walker,
MathernaticalMethods
of
Physics.
Addison-Wesley, Redwood
Misner, Charles W., Kip
S.
Thome, and Jon Archibald Wheeler,
Gravitation.
W.H. Freeman,
Sokolnikoff, Ivan, and R.M. Redheffer,
Mathematics
of
Physics and Modem Engineering.
Taylor, Edwin F., and Jon Archibald Wheeler,
Spacetime Physics.
W.H. Freeman, San Fran-
Thomas, George
B.
Jr., and Ross
L.
Finney,
Calculus andAnalytic Geometry.
Addison-Wesley,
Tolstov, Georgi
P.,
Fourier Series,
translated by Richard A. Silverman. Dover Publications.
Tung, Wu-Ki,
Gmup Theory in Physics.
World Scientific Publishing, Philadelphia,
PA,
1985.
Watson, G.N.,
A Treatise
on
the Theory of Bessel Functions.
Macmillian, New York, 1944.
Wigner, Eugene Paul, Group
Theory and its Applicatons to the Quantum Mechanics
of
Atomic
tions, New
York,
1964.
Hill, New York, 1990.
Wesley, Reading, MA, 1962.
City, CA, 1970.
San Francisco, CA, 1973.
McGraw-Hill, New York, 1966.
cisco, CA, 1963.
Reading, MA, 195
1.
New York, 1962.
Spectra.
Academic
Press,
New York, 1959.
671
This Page Intentionally Left Blank
INDEX
Abelian group, 598,618
Absolutely integrable function, 254
Aliasing, 240-241
Analytic continuation, 175
Analytic function,
see
Complex functions, analytic
Associated Legendre equation,
460
Associated Legendre polynomials, 472
Autocorrelation, 266
Axial vector, 89
Basis functions, 234
Basis vectors
contravariant, 576-579
covariant, 573,576-579
curvilinear coordinates, 49-5
1
nonorthogonal, 564
orthonormal, 2,44,75
position dependent, 655
position independent, 2,564
Bessel
equation,
444
spherical, 354
Bessel functions
first kind, 444446
modified, 453
orthogonality, 448450
second kind,
444-446
spherical, 357
Bilateral Laplace transform,
see
Laplace
transform, double-sided
Block diagonal mamces, 613
Boundary conditions, 342
Dirichlet, 429
for Green’s functions, 384,387-391
homogeneous, 384
Neumann,
429
nonhomogeneous, 387-39 1
Branch cuts,
5
19-522
Branchpoints, 516519
Bromwich contour, 3 13,333
Cartesian coordinates, 2
Cauchy integral formula, 147-150
Cauchy integral theorem, 144147,510-513
Cauchy-Riemann conditions, 141-144,546
Causality, 370
Character, 610-612
Character table, 621-622
Child-Langmuir problem, 359-366,410-412
Christoffel
symbols,
655-659
Class, 610-612.614
Cleverly closed contour, 540
Cofactor, 572
Completeness
C~OSUIE,
175-189,284-285.310-312
Fourier series, 234
group representations, 622-623
Sturm-Liouville eigenfunctions, 437
Completing
the
square,
27 1
Complex functions, 138-202.509-542
analytic,
140-150
derivatives of, 140-144
hyperbolic, 140
logarithm,
140
multivalued functions, 509-542
trigonometric, 139-140
visualization
of,
138
673
674
INDEX
Complex plane, 135-136
Complex variables, 135-137
conjugates, 136
magnitude, 136
polar representation, 136-137
real and imaginary parts, 135, 136
Conductivity tensor, 67-70,76
Conformal mapping,
see
Mapping, conformal
Conservative field, 38
Continuity equation, 27-29,361
Contour deformation, 146-147,532-539,542
Contravariant components, 568-582
Convergence
absolute, 150, 151
complex series, 150-151
Fourier
series,
23 1-234
mean-squared, 232-234
pointwise, 232
ratio test, 151
uniform, 155,232
Fourier transform of, 261-280
Laplace transform of, 320-323
Convolution, 261-265,325
Correlation, 265-266
Cosets, 608-610
Covariant components, 568-570,573-582
Covariant derivatives, 655-659
Cross-correlation, 265
Cross product,
8,
10-12,53,86-92
Curl
Cartesian coordinates,
24
curvilinear coordinates, 55-58
cylindrical coordinates, 58
integral definition, 34
nonorthonormal coordinates, 659
physical picture of, 29-32
spherical coordinates, 58
Current density, 28,68
Curvilinear coordinates, 49-58
Cylindrical coordinates, 45-47
Decomposition of block matrices, 622
Del operator
Cartesian coordinates, 23-24
curvilinear coordinates, 54-58
identities, 32-34.41
Delta function, Dirac,
see
Dirac delta function
Delta, Kronecker,
see
Kronecker delta
DeMoivre’s formula, 203
Density functions, singular, 114-121
Differential equations, 339-403
boundary conditions, 342
constant coefficients, 347-349
coupled, 341
exact differential, 343-345
first-order, 342-346
Fourier transform solutions, 366-371
Green’s function solutions, 376403
homogeneous, 341-342
integrating factor, 345-346
Laplace transform solutions, 371-376
linear, 340-342
nonhomogeneous, 341-342.35 1-354
nonlinear, 340,358-359
order, 341-342
ordinary, 340
partial, 340,391
second-order, 347-354
separation of variables, 342-343,424-475
series solutions, 35&358
terminology, 339-342
Diffusion equation, 391-398
Dipole, 123-125
Dipole moment, 122
Dirac delta function, 100-126
complicated arguments, 108-1 11
derivatives of, 112-1 14
Fourier transform of, 269-270
integral definition, 106-108
integral
of,
11 1-1 12
Laplace transform of,
3
14-3 16
sequence definition, 104-105
shifted arguments, 102
three-dimensional, 115
use
in Green’s function, 378,382
use
in orthogonality relations, 253-254
Direct sum of block matrices, 613
Dirichlet boundary conditions, 429
Discrete Fourier series, 234-242
Displacement vector, 52
Divergence
Cartesian coordinates, 24
curvilinear coordinates, 54-55
cylindrical coordinates, 58
integral definition, 34
nonorthonormal coordinates, 65U59
physical picture of, 27-29
spherical coordinates, 58
nonorthonormal coordinates, 568-570,573
tensors, 70
Double-sided Laplace transform,
see
Laplace
Doublet, 112-1 14
Dyadic product, 7
1
Dot
product, 7-10,53
transform, double-sided
Eigenfunction, 433440
Eigenvalue, 79,435-436
INDEX
675
Eigenvector, 79,435
Einstein summation convention, 2
Electric dipole, 123-125
Electric monopole, 122-123
Electric quadrapole, 133
Elliptical coordinates, 62
Essential singularity, 170
Euler angles, 632
Euler
constant, 445
Euler’s equation, 136 I37
Exact differential, 343-345
Factorial function, 553
Fast Fourier transform
(FFT),
234
Field lines, 20
Fourier series, 219-242
circuit analogy, 223-224
convergence, 23 1-234
discrete form, 234-242
exponential form, 227-23
1
orthogonality conditions, 221-223,228-229
sine-cosine form, 219-223
of
a square wave, 225-227
of a triangular wave, 224-225
Fourier transform, 250-295
circuit analogy, 256.257
of
a convolution, 261
of a cross-correlation, 266
of
a damped sinusoid, 287-290
of a decaying exponential, 283-287
of a delayed function, 258
of a delta function, 269-270
of a derivative, 259-260
differential equation solutions, 366-37
1
of
even and odd functions, 259
existence, 254256
of a Gaussian, 270-273
integral equation solutions, 499
limits of, 303-313
orthogonality condition, 253-254
of a periodic function, 273-275.279-280
of
a product, 260-261
of pure real functions, 259
relation to Fourier series, 250-253.275
of a square pulse, 267-269
transform pair, 252
first kind, 492499
second kind, 492,504
Fredholm equation
Frobenius, method
of,
354358,444, 461,
469
Gamma function, 553-554
Gauss’s theorem, 35-36
Gaussian
Fourier transform of, 270-273
sequence function,
105
General relativity, 565-566
Generalized coordinates,
see
Curvilinear
Generalized functions, 100, 103
Geodesics,
566
Gibbs phenomenon, 233
Gradient
coordinates
Cartesian coordinates, 24
curvilinear coordinates, 54
cylindrical coordinates, 58
integral definition, 34
nonorthonormal coordinates, 582,656
physical picture of, 24-27
spherical coordinates, 58
Gram-Schmidt orthogonalization, 437
Green’s functions, 376403,647-651
boundary conditions, 384,387-391
for diffusion equation, 391-398
for
driven wave equation, 39843
multiple independent variables, 391-403
for a stretched string, 385-391
symmetry properties, 398,422423
translational invariance, 378,398
Green’s theorem, 3637
Groups, 597-634
Abelian, 598,618
character, 610-612
character table, 621-622
class, 610-612, 614
continuous, 630-634
cosets,
608-610
D2.601403
D3.604-605
definition, 597-598
finite, 598-607
multiplication table, 598-599
order, 598
permutation, 607
rearrangement lemma, 599
representations,
see
Representations, group
rotation, 630-633
SU(2). 633-634
subgroups, 607-608
C,,
598-601
Harmonic oscillator
damped, driven, 368-37 1
simple, 347
undamped, driven, 378-381
Heaviside step function, 11 1-1 12, 316-317
Heisenberg uncertainty principle, 273