Table 7.2 Examples of slow functions
sð _
uÞ and their corresponding smoothed images
~
sð_ zÞ for
u(s) =
− asin(s). Lengthy expressions for
~
s
c1 and
~
s
c2 in
(c) and (d) are omitted; they are smooth functions with smooth transitions at the boundaries of their intervals of definition
sð _
uÞ
~
sð_ zÞ
s _
z þ
u
0
ðsÞ
ð
Þ
h
i
; u
0
¼ Àa cos
s
s (—),
~
s (–
– –)
(a)
sgnð
_
uÞ
1
À
2
p arccosð_
z=aÞ; j_
zj a
sgnð_
zÞ;
j_
zj [ a
&
s,s ~
u,z
. .
-a
a
(b)
sgnð
_
uÞ À
_
u þ
1
3
_
u
3
1
À
2
p arccosð_
z=aÞ þ ð
1
2 a
2
À 1Þ_
z þ
1
3 _
z
3
for
_
z
j j
a
sgnð_
zÞ þ ð
1
2 a
2
À 1Þ_
z þ
1
3 _
z
3
for
_
z
j j
[ a
&
(c)
_
u;
j
_
uj 1
sgnð
_
uÞ; j
_
uj [ 1
&
_
z;
~
s
c1 ð_
zÞ;
sgnð_
zÞ;
j_
zj 1
À
a
j_
zj 2 ½1 À
a; 1
þ
aŠ
j_
zj [ 1
þ
a
8
<
:
(continued)
422
7 Special Effects of High-Frequency Excitation