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In-Depth Information
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
=
u
:=
F
(
u
):=
x
0
1
√
1
−x
2
,
d
x
u
:= arcsin
x ⇒
√
1
x
2
=arcsin
x.
(C.1)
u
2
=
1
1
−x
2
,
−
End of Example.
C-2 Elliptic Kernel, Elliptic Modulus, Elliptic Functions, Elliptic
Integrals
Such a well-known formula from circul
ar trig
onometry 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:
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