x
y
Figure 10.1: Sr (x) and Ss (y) when 0 < s < r − ρ(x, y).
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
x6
x7
s
x4
a
r
x1
x5
x2
x3
Figure 10.2: The minimum distance from
a to an element of L = {x1, ... , x7} is r.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Localized wave
0.4
f(t)
0.2
0.0
-0.2
-0.4
-1.0
-0.5
0.0
0.5
1.0
t
Figure 10.3: Plot of the localized wave defined by the function
f(t) = exp − (t + 1)−2(t − 1)−2 sin ωt if −1 < t < 1 and
f(t) = 0 for all other values of . The support of is the closed set
[−1, 1].
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
∂ Sr (x )
Sr (x )
S \ Sr (x )
x
r
Figure 10.4: Sr(x), S \Sr(x) and ∂Sr(x) in S ⊂ R2
under the 2-norm. In this example, but not in all
metric spaces, Sr(x) = Sr(x).
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
r3
r1
y2
y1
y3
r2
Figure 10.5: Illustration
of the construction of y1, y2 and y3.
The sequence y1, y2, . . . is unbounded.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
0.0010
Error of economized series vs. x
error
0.0005
0.0000
-0.0005
-0.0010
0.00
0.25
0.50
x
0.75
1.00
Figure 10.6: Error in the economized sine series, Eq. (10.317).
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Square wave
2
f(x)
1
0
-1
-2
-4
-2
0
2
4
x
Figure 10.7: Periodic square wave of unit amplitude.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
40
20
0
-20
-40
-10
-8
-6
-4
-2
0
2
4
6
8
10
x
Figure 10.8: Three periods of the function (10.391) for n = 3, showing
that at odd multiples of π the function has jump discontinuities, but its
derivatives are continuous.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Fourier sum (11 terms)
2
s11 (x)
1
0
-1
-2
-4
-2
0
2
4
x
Figure 10.9: Partial sum of the Fourier series (10.388) for k = 0 through k = 5.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Fourier sum (41 terms)
2
s41 (x)
1
0
-1
-2
-4
-2
0
2
4
x
Figure 10.10: Partial sum of the Fourier series (10.388) for k = 0 through k = 20.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Legendre sum (11 terms)
2
s11 (x)
1
0
-1
-2
-1.0
-0.5
0.0
0.5
1.0
x
Figure 10.11: Partial sum of the Legendre series (10.416) for k = 0 through k = 5.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Legendre sum (41 terms)
2
s41 (x)
1
0
-1
-2
-1.0
-0.5
0.0
0.5
1.0
x
Figure 10.12: Partial sum of the Legendre series (10.416) for k = 0 through k = 20.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press