8-2 Special mapping equations 231
Corollary 8.5 (Power series (1 + x)
y , reformulation 3).
(1 − E
2 cos
2 ∆)
−3/2 =
= 1 +
3
4
E
2 +
45
64
E
4 +
175
256
E
6 + O 1 (E
8 )+
+
3
4
E
2 +
15
16
E
4 +
525
512
E
6 + O 2 (E
8 )
cos 2∆+
+
15
64
E
4 +
105
256
E
6 + O 3 (E
8 )
cos 4∆ +
35
512
E
6 + O 4 (E
8 )
cos 6∆ + O 5 (E
8 ) .
(8.36)
End of Corollary.
Corollary 8.6 (Power series (1 + x)
y , multiplication with (1 − E
2 )).
(1 − E
2 )(1 − E
2 cos
2 ∆)
−3/2 =
= 1 −
1
4
E
2
−
3
64
E
4
−
5
256
E
6 + O 1 (E
8 )+
+
3
4
E
2 +
3
16
E
4 +
45
512
E
6 + O 2 (E
8 )
cos 2∆+
+
15
64
E
4 +
45
256
E
6 + O 3 (E
8 )
cos 4∆ +
35
512
E
6 + O 4 (E
8 )
cos 6∆ + O 5 (E
8 ) .
(8.37)
End of Corollary.
Corollary 8.7 (Termwise integration of uniformly convergent series).
cos nxdx =
1
n
sin nx ∀ n ∈ Z ,
∆
0
(1 − E
2 )(1 − E
2 cos
2 ∆
)
−3/2 d∆
=
=
1 −
1
4
E
2
−
3
64
E
4
−
5
256
E
6 + O 1 (E
8 )
∆+
+
3
8
E
2 +
3
32
E
4 +
45
1024
E
6 + O 2 (E
8 )
sin 2∆+
+
15
256
E
4 +
45
1024
E
6 + O 3 (E
8 )
sin 4∆ +
35
3072
E
6 + O 4 (E
8 )
sin 6∆ + O 5 (E
8 ) .
(8.38)
End of Corollary.