Geography Reference
In-Depth Information
(
a
3
= 0 admitted), we arrive at elliptic integrals whose kernel is a square root of a
polynomial up to degree four, with distinct nodal points. Definition
C.7
is a collective summary
of elliptic integrals of the first, second, and third kinds given in both polynomial form and trigono-
metric form. In general, integrals of the following form are needed:
I
=
R
x,
a
J
x
J
+
a
J−
1
x
J−
1
+
...
+
a
1
x
+
a
0
d
x
(
J ≥
5)
.
=0
,a
4
(C.32)
Definition C.7 (Elliptic integrals of the first, second, and third kinds).
The following integrals
F
(
x, k
)
,E
(
x, k
), and
π
(
x, k
) are called
normal elliptic integrals
of the
first, the second, and the third kind:
F
(
x, k
):=
x
0
d
x
(1
k
2
x
2
)
,
−
x
2
)(1
−
√
1
−
k
2
x
2
d
x
(1
− x
2
)
E
(
x, k
):=
x
0
,
(C.33)
π
(
x, k
):=
x
0
d
x
(1 +
nx
2
)
(1
k
2
x
2
)
.
−
x
2
)(1
−
The following integrals
F
(
φ, k
)
,E
(
φ, k
),and
π
(
φ, k
) are called
trigonometric elliptic integrals
of the first, the second, and the third kind:
F
(
φ, k
):=
φ
0
d
φ
1
,
k
2
sin
2
φ
−
E
(
φ, k
):=
φ
0
1
− k
2
sin
2
φ
d
φ
=
u
0
dn
2
(u
,
k)du
,
(C.34)
π
(
φ, k
):=
φ
0
d
φ
1+
n
sin
2
φ
1+
k
2
sin
2
φ
.
Note that for
φ
=
π/
2
,
i
.
e
.F
(
π/
2
,k
)
,E
(
π/
2
,k
), and
π
(
π/
2
,k
), these trigonometric elliptic
integrals are called
complete
.
End of Definition.
For the numerical analysis of elliptic functions, the periodicity of sn
u
,cn
u
, and dn
u
is of focal
interest. Lemma
C.8
defines the periodic properties of sn
u
,cn
u
, and dn
u
more precisely. Note
that a proof can be based upon the addition theorem of elliptic functions.
Lemma C.8 (Periodicity of elliptic functions: C. G. J. Jacobi).
The elliptic functions
u
,cn
u
, and dn
u
are doubly periodic in the sense of (
n, m
∈
{
1
,
2
,
3
,
4
,
5
...}
)
sn(
u
+4
mK
+2
n
i
K
)=sn
u,
cn[
u
+4
mK
+2
n
(
K
+i
K
)] = cn
u,
(C.35)
dn(
u
+2
mK
+4
n
i
K
)=dn
u,
Search WWH ::
Custom Search