
30.14. Elliptic Functions 1549
◮ Double-argument formulas.
sn(2u) =
2 sn u cn u dn u
1 −k
2
sn
4
u
=
2 sn u cn u dn u
cn
2
u + sn
2
u dn
2
u
,
cn(2u) =
cn
2
u − sn
2
u dn
2
u
1 −k
2
sn
4
u
=
cn
2
u − sn
2
u dn
2
u
cn
2
u + sn
2
u dn
2
u
,
dn(2u) =
dn
2
u − k
2
sn
2
u cn
2
u
1 −k
2
sn
4
u
=
dn
2
u + cn
2
u (dn
2
u −1)
dn
2
u − cn
2
u (dn
2
u −1)
.
◮ Half-argument formulas.
sn
2
u
2
=
1
k
2
1 −dn u
1 + cn u
=
1 − cn u
1 + dn u
,
cn
2
u
2
=
cn u + dn u
1 + dn u
=
1 −k
2
k
2
1 −dn u
dn u − cn u
,
dn
2
u
2
=
cn u + dn u
1 + cn u
= (1 −k
2
)
1 − cn u
dn u −cn u
.
◮ Argument addition formulas.
sn(u ± v) =
sn u cn v dn v ± sn v cn u dn u
1 − k
2
sn
2
u sn
2
v
,
cn(u ± v) =
cn u cn v ∓ sn u sn v dn u dn v
1 − k
2
sn
2
u sn
2
v
,
dn(u ± v) =
dn u dn v ∓ k
2
sn u sn v cn u cn v
1 −k
2
sn
2
u sn
2
v
.
◮