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In-Depth Information
The standard elliptic functions satisfy special identities collected in Corollary
C.4
,where1
−k
2
is the
complementary elliptic modulus
. Similarly, we summarize addition formulae of elliptic func-
tions in Corollary
C.5
.
Corollary C.4 (Special identities).
sn
2
u
+cn
2
u
=1
,k
2
sn
2
u
+dn
2
u
=1
,
(C.29)
k
2
cn
2
u
+(1
k
2
)=dn
2
u,
cn
2
u
+(1
k
2
)sn
2
u
=dn
2
u.
−
−
End of Corollary.
Corollary C.5 (Addition formulae).
sn(
u
+
v
)=
sn
u
cn
v
dn
v
+sn
v
sn
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
−
,
(C.30)
k
2
sn
u
sn
v
cn
u
cn
v
dn(
u
+
v
)=
dn
u
dn
v
−
.
1
−
k
2
sn
2
u
sn
2
v
End of Corollary.
Series expansions of “sinus amplitudinis”, “cosinus amplitudinis”, and “delta amplitudinis” are
useful in Mathematical Cartography. In the following Corollary
C.6
, series expansions of “sinus
amplitudinis”, “cosinus amplitudinis”, and “delta amplitudinis” are summarized.
Corollary C.6 (Series expansions of elliptic functions).
1
1
6
(1 +
k
2
)
u
3
+
120
(1 + 14
k
2
+
k
4
)
u
5
+O(
u
7
)
,
sn (
u,k
)=
u
−
1
2
u
2
+
1
1
4
k
2
)
u
4
720
(1 + 44
k
2
+16
k
4
)
u
6
+O(
u
8
)
,
cn (
u,k
)=1
−
24
(1
−
−
(C.31)
1
2
k
2
u
2
+
1
1
24
(4
k
2
+
k
4
)
u
4
720
(16
k
2
+44
k
3
+
k
6
)
u
6
+O(
u
8
)
.
dn (
u,k
)=1
−
−
End of Corollary.
In analysis, we have been made familiar with integrals of type
R
(
x,
√
ax
2
+2
bx
+
c
)d
x
whose ker
nel is
a square root of a polynomial up to degree two like
d
x/x,
d
x/
(1 +
x
2
),
or
d
x/
√
1
−
x
2
. Those integrals can be integrated, for exa
mple, to ln
x
, arctan
x
,ora
rc-
sin
x
. Alternatively, if the integral is of the form
R
(
x,
√
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
)d
x
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