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( 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,
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