C-2 Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals 521
Proof (Proof of formulae (C.8) and (C.9)).
x := sn u ,
1 − x 2 = cn u ,
1 − k 2 x 2 = dn u ,
(C.20)
dx
du
=
1
du/dx
=
(1 − x 2 ) (1 − k 2 x 2 ) ,
dx
du
= cn u dn u q. e. d.
dx
du
2
=
1 − x
2
1 − k
2 x
2
q. e. d.
(C.21)
End of Proof (Proof of formulae (C.8) and (C.9)).
Proof (Proof of formulae (C.10) and (C.11)).
y := cn u =
1 − x 2 , sn u =
1 − y 2 = x , dn u =
1 − k 2 (1 − y 2 ) =
1 − k 2 x 2 , (C.22)
dy
du
=
d
√
1 − x 2
du
= −
x
√
1 − x 2
dx
du
,
dy
du
= −x
1 − k 2 x 2 = −sn u dn u q. e. d.
dy
du
2
=
1 − y
2
1 − k
2 + k
2 y
2
q. e. d.
(C.23)
End of Proof (Proof of formulae (C.10) and (C.11)).
Proof (Proof of formulae (C.12) and (C.13)).
z := dn u =
1 − k 2 x 2 , sn u =
√
1 − z 2
k
= x , cn u =
k 2 − (1 − z 2 )
k
=
1 − x 2 , (C.24)
dz
du
=
d
√
1 − k 2 x 2
du
= −
k
2 x
√
1 − k 2 x 2
dx
du
,
dz
du
= −k
2 sn u cn u q. e. d.
dz
du
= −k
2 x
1 − x 2 = −
1 − z 2
k 2 − (1 − z 2 ) ,
dz
du
2
=
1 − z
2
z
2
−
1 − k
2
q. e. d.
(C.25)
End of Proof (Proof of formulae (C.12) and (C.13)).