C Elliptic integrals
Elliptic kernel, elliptic modulus, elliptic functions, and elliptic integrals. Differential equations of elliptic
functions. Sinus amplitudinis, cosinus amplitudinis, and delta amplitudinis.
We experience elliptic integrals when we are trying to compute the length of a meridian arc or the
length of a geodesic of an ellipsoid-of-revolution. Here, we begin with an interesting example from
circular trigonometry, which is leading us to the notion of elliptic integrals of the first kind as well as
elliptic functions.
C-1 Introductory example
Example C.1 (Elliptic functions).
u := arcsin x ⇒
⎡
⎢
⎢
⎣
u
=
1
√
1 − x 2
,
u
2 =
1
1 − x 2 ,
u := F (x) :=
x
0
dx
1 − x 2
= arcsin x .
(C.1)
End of Example.
C-2 Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals
Such a well-known formula from circular trigonometry expresses the inverse function of sin x as an
integral. The integral kernel is 1/
√
1 − x 2 . As soon as we switch to elliptic trigonometry, the following
“elliptic kernel” appears:
1
√
1 − x 2
1
√
1 − k 2 x 2
(0 ≤ k
2
≤ 1) .
(C.2)
We here note that the factor k is called the elliptic modulus and that the function x = am (u, k) is
called Jacobian amplitude within
u := F (x, k) :=
x
0
dx
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
dφ
1 − k 2 sin
2 φ
, φ = F
−1 (u, k) = am (u, k) .
(C.4)
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) ,
cos φ = cos am(u, k) =: cn (u, k) ,
(C.5)
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.
Elliptic kernel, elliptic modulus, elliptic functions, and elliptic integrals. Differential equations of elliptic
functions. Sinus amplitudinis, cosinus amplitudinis, and delta amplitudinis.
We experience elliptic integrals when we are trying to compute the length of a meridian arc or the
length of a geodesic of an ellipsoid-of-revolution. Here, we begin with an interesting example from
circular trigonometry, which is leading us to the notion of elliptic integrals of the first kind as well as
elliptic functions.
C-1 Introductory example
Example C.1 (Elliptic functions).
u := arcsin x ⇒
⎡
⎢
⎢
⎣
u
=
1
√
1 − x 2
,
u
2 =
1
1 − x 2 ,
u := F (x) :=
x
0
dx
1 − x 2
= arcsin x .
(C.1)
End of Example.
C-2 Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals
Such a well-known formula from circular trigonometry expresses the inverse function of sin x as an
integral. The integral kernel is 1/
√
1 − x 2 . As soon as we switch to elliptic trigonometry, the following
“elliptic kernel” appears:
1
√
1 − x 2
1
√
1 − k 2 x 2
(0 ≤ k
2
≤ 1) .
(C.2)
We here note that the factor k is called the elliptic modulus and that the function x = am (u, k) is
called Jacobian amplitude within
u := F (x, k) :=
x
0
dx
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
dφ
1 − k 2 sin
2 φ
, φ = F
−1 (u, k) = am (u, k) .
(C.4)
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) ,
cos φ = cos am(u, k) =: cn (u, k) ,
(C.5)
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.