Geography Reference
In-Depth Information
1
1
√
1
√
1
k
2
(0
≤
≤
1)
.
(C.2)
−
x
2
−
k
2
x
2
We here note that the factor
k
is called the
elliptic modulus
and that the function
x
=am(
u,k
)
called
Jacobian amplitude
within
u
:=
F
(
x, k
):=
x
0
d
x
√
1
− x
2
√
1
− k
2
x
2
,x
=
F
−
1
(
u,k
)=am(
u,k
)
.
(C.3)
By means of the elementary substitution
x
=sin
Φ
, the elliptic integral of the first kind is reduced
to the following normal trigonometric form:
u
:=
F
(
φ, k
):=
φ
0
d
φ
,φ
=
F
−
1
(
u,k
)=am(
u,k
)
.
1
(C.4)
k
2
sin
2
φ
−
We here additionally note that C. G. J. Jacobi (1804-1851) and N. H. Abel (1802-1829) had the
bright idea to replace Legendre's elliptic integral of the first kind by its inverse function. The
inverse function is the “simplest elliptic function” : see Definition
C.1
.
Definition C.1 (Elliptic functions).
sin
φ
= sin am(
u,k
)=:sn(
u,k
)
,
(C.5)
cos
φ
= cosam(
u,k
)=:cn(
u,k
)
,
1
− k
2
sin
2
φ
=
1
− k
2
sin
2
am(
u,k
)=:dn(
u,k
)
,
(C.6)
to be read “sinus amplitudinis”, “cosinus amplitudinis”, and “delta amplitudinis”.
End of Definition.
These functions are doubly periodic generalizations of the circular trigonometric functions satis-
fying
sn(
u,
0) = sin
u,
cn(
u,
0) = cos
u,
(C.7)
dn(
u,
0) = 1
.
sn(
u,k
)
,
cn(
u,k
), and dn(
u,k
) may also be defined as solutions of the differential equa-
tions (
C.9
), (
C.11
), and (
C.13
) of first order or as solutions of the differential equa-
tions (
C.15
), (
C.17
), and (
C.19
) of second order: see Lemmas
C.2
and
C.3
.
Lemma C.2 (Differential equations of elliptic functions).
The elliptic functions sn(
u,k
), cn (
u,k
),anddn(
u,k
) satisfy the trigonometric and the algebraic
differential equations of first order that follow:
x
=sn
u,
sn
u
=
dsn
u
d
u
=cn
u
dn
u,
(C.8)
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