H-5 The inverse of a special univariate homogeneous polynomial 619
Fourth term (see (H.14), (H.30)):
t 4 = t 4 (A
∗ , B
∗ ) :=
1 − 2E
2 cos
2 A
∗ + E
4 cos
2 A
∗
E sin A ∗
arcsin
E sin A
∗
√
1 − 2E 2 cos 2 A ∗ + E 4 cos 2 A ∗
, (H.111)
t 4 = t 4 (Λ
∗ , Φ
∗ ) :=
1 − E
2
E sin Φ ∗ ×
×
1 − sin
2 (Λ
∗
− Ω) cos
2 Φ
∗
cos 2 (Λ ∗ − Ω) cos 2 Φ ∗ + (1 − E 2 ) 2 sin
2 Φ ∗
arcsin
E sin Φ
∗
1 − sin
2 (Λ ∗ − Ω) cos 2 Φ ∗
.
(H.112)
H-5 The inverse of a special univariate homogeneous polynomial
In order to solve (H.65) for sin Φ
∗ , we proceed to present the series expansions of ar tanh x in (H.113)
and of (1 + x)
−1 in (H.114) (compare with M. Abramowitz and J. A. Stegun 1965). Those series expansions is applied to the two terms (H.115) and (H.116) which appear in (H.65). In particular,
we recognize the homogeneous polynomial form of (H.117) as soon as we substitute x := sin Φ
∗ by
(H.118) and y by (H.119). The inverse of the univariate homogeneous polynomial (H.120) represented
by (H.121) is computed up to degree five. Forward and backward substitution amount to (H.129)
reevaluated by means of the final solution to the inversion by (H.131).
ar tanh x = x +
x
3
3
+
x
5
5
+
x
7
7
+ O(x
9 ) (|x| < 1) ,
(H.113)
(1 + x)
−1 = 1 − x + x
2
− x
3 + O(x
4 ) (−1 < x < 1) ,
(H.114)
1
2E
ar tanh (E sin Φ
∗ ) =
1
2
sin Φ
∗ +
1
6
E
2 sin
3 Φ
∗ +
1
10
E
4 sin
5 Φ
∗ + O(E
6 ) ,
(H.115)
sin Φ
∗
2(1 − E 2 sin
2 Φ ∗ )
=
1
2
sin Φ
∗ +
1
2
E
2 sin
3 Φ
∗ +
1
2
E
4 sin
5 Φ
∗ + O(E
6 ) ,
(H.116)
1
2E
ar tanh (E sin Φ
∗ ) +
sin Φ
∗
2(1 − E 2 sin
2 Φ ∗ )
= sin Φ
∗ +
2
3
E
2 sin
3 Φ
∗ +
3
5
E
4 sin
5 Φ
∗ + O(E
6 ) . (H.117)
(H.65) can now be written as univariate special homogeneous polynomial of degree n, namely
x := sin Φ
∗ ,
(H.118)
y := c 4
1
2E
ar tanh (E sin Φ) +
sin Φ
2(1 − E 2 sin
2 Φ)
=
= c 4
sin Φ +
2
3
E
2 sin
3 Φ +
3
5
E
4 sin
5 Φ + O(E
6 )
,
(H.119)
y(x) = a 11 x + a 13 x
3 + a 15 x
5 + · · · + a 1n x
n (n odd) ,
(H.120)
x(y) = b 11 y + b 13 y
3 + b 15 y
5 + · · · + b 1n y
n
(n odd) ,
(H.121)
subject to
a 11 = 1 .
(H.122)