522 C Elliptic integrals
Proof (Proof of formula (C.15)).
x
= sn
u = cn u dn u
⇒
x
= sn
u = cn
u dn u + cn u dn
u ,
x
= −sn u dn
2 u − k
2 sn u cn
2 u ,
x
= −x
1 − k
2 x
2
− k
2 x
1 − x
2
= −x + k
2 x
3
− k
2 x + k
2 x
3 ,
x
= −x
1 + k
2
+ 2k
2 x
3
q. e. d.
(C.26)
End of Proof (Proof of formula (C.15)).
Proof (Proof of formula (C.17)).
y
= cn
u = −sn u dn u
⇒
y
= cn
u = −sn
u dn u − sn u dn
u ,
y
= −cn u dn
2 u + k
2 sn
2 u cn u ,
y
= −y
1 − k
2
1 − y
2
+ k
2 y
1 − y
2
= −y + k
2 y
1 − y
2
+ k
2 y
1 − y
2
,
y
= −y + 2k
2 y
1 − y
2
= −y + 2k
2 y − 2k
2 y
3 ,
y
= −y
1 − 2k
2
− 2k
2 y
3
q. e. d.
(C.27)
End of Proof (Proof of formula (C.17)).
Proof (Proof of formula (C.19)).
z
= dn
u = −k
2 sn u cn u
⇒
z
= −k
2 sn
u cn u − k
2 sn u cn
u ,
z
= −k
2 cn
2 u dn u + k
2 sn
2 u dn u ,
z
= −
k
2
−
1 − z
2
z +
1 − z
2
z = −k
2 z + 2z
1 − z
2
,
z
= +
2 − k
2
z − 2z
3 q. e. d.
(C.28)
End of Proof (Proof of formula (C.19)).