230 8 “Ellipsoid-of-revolution to tangential plane”
Corollary 8.1 (Power series (1 + x)
y , Taylor expansion).
(1 + x)
y =
= 1 +
1
1!
y[1 + x]
y−1
x=0 x
1 +
1
2!
y(y − 1)[1 + x]
y−2
x=0 x
2 +
+
1
3!
y(y − 1)(y − 2)[1 + x]
y−3
x=0 x
3 + O(x
4 )
∀ |x| < 1 ,
(1 + x)
y := (1 − E
2 cos
2 ∆)
−3/2 , x := −E
2 cos
2 ∆ , y := −
3
2
.
(8.32)
End of Corollary.
Corollary 8.2 (Power series (1 + x)
y , reformulation 1).
(1 − E
2 cos
2 ∆)
−3/2 =
= 1 +
3
2
E
2 cos
2 ∆ +
3 · 5
2 · 4
E
4 cos
4 ∆ +
3 · 5 · 7
2 · 4 · 6
E
6 cos
6 ∆ + O(E
8 ) .
(8.33)
End of Corollary.
Corollary 8.3 (Power series (1 + x)
y , cosine powers).
cos
2 ∆ =
1
2
+
1
2
cos 2∆ ,
cos
4 ∆ =
3
8
+
1
2
cos 2∆ +
1
8
cos 4∆ ,
cos
6 ∆ =
5
16
+
15
32
cos 2∆ +
3
16
cos 4∆ +
1
32
cos 6∆ .
(8.34)
End of Corollary.
Corollary 8.4 (Power series (1 + x)
y , reformulation 2).
(1 − E
2 cos
2 ∆)
−3/2 =
= 1 +
3
2
E
2
1
2
+
1
2
cos 2∆
+
+
15
8
E
4
3
8
+
1
2
cos 2∆ +
1
8
cos 4∆
+
+
35
16
E
6
5
16
+
15
32
cos 2∆ +
3
16
cos 4∆ +
1
32
cos 6∆
+
+O(E
8 ) .
(8.35)
End of Corollary.