The argument k is called the elliptic modulus k2... Incomplete Elliptic Integrals Elliptic Integrals 18.13.2-1... The quantity k is called the elliptic modulus k2... Elliptic Functions A
Trang 118.13 Elliptic Integrals
18.13.1 Complete Elliptic Integrals
18.13.1-1 Definitions Properties Conversion formulas
Complete elliptic integral of the first kind:
K(k) =
π/2 0
dα
√
1– k2sin2α =
1 0
dx
(1– x2)(1– k2x2).
Complete elliptic integral of the second kind:
E(k) =
π/2 0
√
1– k2sin2α dα=
1 0
√
1– k2x2
√
1– x2 dx.
The argument k is called the elliptic modulus (k2<1)
Notation:
k =√
1– k2, K (k) =K(k ), E (k) =E(k ),
where k is the complementary modulus.
Properties:
K(–k) =K(k), E(–k) =E(k);
K(k) =K (k ), E(k) =E (k );
E(k)K (k) +E (k)K(k) –K(k)K (k) = π
2.
Conversion formulas for complete elliptic integrals:
K
1
– k
1+ k
= 1+ k
2 K(k), E
1
– k
1+ k
1+ k
E(k) + k K(k)
, K
2√ k
1+ k
= (1+ k)K(k),
E
2√ k
1+ k
1+ k
2E(k) – (k )2K(k)
18.13.1-2 Representation of complete elliptic integrals in series form
Representation of complete elliptic integrals in the form of series in powers of the modulus k:
K(k) = π
2
1+
1
2
2
k2+1 × 3
2 × 4
2
k4+· · · +
(2n–1)!!
(2n)!!
2
k2n+· · ·
,
E(k) = π
2
1–
1
2
2
k2
1 –
1 × 3
2 × 4
2
k4
3 –· · · –
(2n–1)!!
(2n)!!
2
k2n
2n–1 –· · ·
Trang 2
970 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
Representation of complete elliptic integrals in the form of series in powers of the
comple-mentary modulus k =√
1– k2:
K(k) = π
1+ k
1 +
1
2
21
– k
1+ k
2
+
1 × 3
2 × 4
21
– k
1+ k
4
+· · · +
( 2n– 1 )!!
( 2n)!!
21
– k
1+ k
2n
+· · ·
,
K(k) = ln 4
k +
1 2
2
ln 4
k – 2
1 × 2
(k )2+
1 × 3
2 × 4
2
ln 4
k – 2
1 × 2–
2
3 × 4
(k )4 +
1 × 3 × 5
2 × 4 × 6
2
ln 4
k – 2
1 × 2–
2
3 × 4–
2
5 × 6
(k )6+· · · ;
E(k) = π(1+ k )
4
1 + 1
2 2 –
1
– k
1+ k
2
+ 1 2
( 2 × 4 )2
1
– k
1+ k
4
+· · · +
( 2n– 3 )!!
( 2n)!!
21
– k
1+ k
2n
+· · ·
,
E(k) =1 + 1
2
ln 4
k – 1
1 × 2
(k )2+ 1 2 × 3
2 2 × 4
ln 4
k – 2
1 × 2–
1
3 × 4
(k )4 + 1 2 × 3 2 × 5
2 2 × 4 2 × 6
ln 4
k – 2
1 × 2–
2
3 × 4–
1
5 × 6
(k )6+· · ·
18.13.1-3 Differentiation formulas Differential equations
Differentiation formulas:
dK(k)
dk = E(k)
k(k )2 –
K(k)
k , dE(k)
dk = E(k) –K(k)
The functionsK(k) andK (k) satisfy the second-order linear ordinary differential
equa-tion
d dk
k(1– k2)dK
dk
– kK=0
The functionsE(k) andE (k) –K (k) satisfy the second-order linear ordinary differential
equation
(1– k2) d
dk
k dE dk
+ kE=0
18.13.2 Incomplete Elliptic Integrals (Elliptic Integrals)
18.13.2-1 Definitions Properties
Elliptic integral of the first kind:
F(ϕ, k) =
ϕ 0
dα
√
1– k2sin2α =
sinϕ 0
dx
(1– x2)(1– k2x2).
Elliptic integral of the second kind:
E(ϕ, k) =
ϕ 0
√
1– k2sin2α dα=
sinϕ 0
√
1– k2x2
√
1– x2 dx.
Trang 3Elliptic integral of the third kind:
Π(ϕ, n, k) =
ϕ
0
dα
(1– n sin2α) √
1– k2sin2α =
sinϕ 0
dx
(1– nx2)
(1– x2)(1– k2x2).
The quantity k is called the elliptic modulus (k2<1), k =√
1– k2is the complementary
modulus, and n is the characteristic parameter.
Complete elliptic integrals:
K(k) = F
π
2, k
, E(k) = E
π
2, k
,
K (k) = Fπ
2, k
, E (k) = Eπ
2, k
Properties of elliptic integrals:
F(–ϕ, k) = –F (ϕ, k), F (nπ ϕ, k) =2nK(k) F (ϕ, k);
E(–ϕ, k) = –E(ϕ, k), E(nπ ϕ, k) =2nE(k) E(ϕ, k).
18.13.2-2 Conversion formulas
Conversion formulas for elliptic integrals (first set):
F
ψ, 1
k
= kF (ϕ, k), E
ψ, 1
k
= 1
k
E(ϕ, k) – (k )2F (ϕ, k)
,
where the angles ϕ and ψ are related by sin ψ = k sin ϕ, cos ψ =
1– k2sin2ϕ.
Conversion formulas for elliptic integrals (second set):
F
ψ, 1– k
1+ k
= (1+ k )F (ϕ, k), E
ψ, 1– k
1+ k
1+ k
E(ϕ, k) + k F (ϕ, k)
– 1– k
1+ k sin ψ, where the angles ϕ and ψ are related by tan(ψ – ϕ) = k tan ϕ.
Transformation formulas for elliptic integrals (third set):
F
ψ, 2√ k
1+ k
= (1+ k)F (ϕ, k), E
ψ, 2√ k
1+ k
1+ k
2E(ϕ, k) – (k )2F (ϕ, k) +2k sin ϕ cos ϕ
1+ k sin2ϕ
1– k2sin2ϕ
,
where the angles ϕ and ψ are related by sin ψ = (1+ k) sin ϕ
1+ k sin2ϕ
Trang 4972 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
18.13.2-3 Trigonometric expansions
Trigonometric expansions for small k and ϕ:
F(ϕ, k) = 2
π K(k)ϕ – sin ϕ cos ϕ
a0+ 2
3a1sin2ϕ+
2 × 4
3 × 5a2sin4ϕ+· · ·
,
a0= 2
π K(k) –1, a n = a n–1–
(2n–1)!!
(2n)!!
2
k2n;
E(ϕ, k) = 2
π E(k)ϕ – sin ϕ cos ϕ
b0+ 2
3b1sin2ϕ+
2 × 4
3 × 5b2sin4ϕ+· · ·
,
b0=1– 2
π E(k), b n = b n–1–
(2n–1)!!
(2n)!!
2
k2n
2n–1.
Trigonometric expansions for k →1:
F (ϕ, k) = 2
π K
(k) ln tan
ϕ
2 +
π
4
– tan ϕ
cos ϕ
a
0– 2
3a 1tan2ϕ+
2 × 4
3 × 5a 2tan4ϕ–· · ·
,
a
0 = 2
π K
(k) –1, a
n = a n–1–
(2n–1)!!
(2n)!!
2
(k )2n;
E(ϕ, k) = 2
π E
(k) ln tan
ϕ
2 +
π
4
+ tan ϕ
cos ϕ
b
0– 2
3b 1tan2ϕ+
2 × 4
3 × 5b 2tan4ϕ–· · ·
,
b
0= 2
π E
(k) –1, b
n = b n–1–
(2n–1)!!
(2n)!!
2
(k )2n
2n–1.
18.14 Elliptic Functions
An elliptic function is a function that is the inverse of an elliptic integral An elliptic function
is a doubly periodic meromorphic function of a complex variable All its periods can be written in the form 2mω1+2nω2 with integer m and n, where ω1 and ω2 are a pair of
(primitive) half-periods The ratio τ = ω2/ω1is a complex quantity that may be considered
to have a positive imaginary part, Im τ >0
Throughout the rest of this section, the following brief notation will be used: K=K(k)
and K =K(k ) are complete elliptic integrals with k =√
1– k2
18.14.1 Jacobi Elliptic Functions
18.14.1-1 Definitions Simple properties Special cases
When the upper limit ϕ of the incomplete elliptic integral of the first kind
u=
ϕ 0
dα
√
1– k2sin2α = F (ϕ, k)
is treated as a function of u, the following notation is used:
u = am ϕ.
Trang 5Naming: ϕ is the amplitude and u is the argument.
Jacobi elliptic functions:
sn u = sin ϕ = sin am u (sine amplitude),
cn u = cos ϕ = cos am u (cosine amplitude),
dn u =
1– k2sin2ϕ= dϕ
du (delta amlplitude).
Along with the brief notations sn u, cn u, dn u, the respective full notations are also used: sn(u, k), cn(u, k), dn(u, k).
Simple properties:
sn(–u) = – sn u, cn(–u) = cn u, dn(–u) = dn u;
sn2u+ cn2u=1, k2sn2u+ dn2u=1, dn2u – k2cn2u=1– k2,
where i2= –1
Jacobi functions for special values of the modulus (k =0and k =1):
sn(u,0) = sin u, cn(u,0) = cos u, dn(u,0) =1;
sn(u,1) = tanh u, cn(u,1) = 1
cosh u, dn(u,1) = 1
cosh u.
Jacobi functions for special values of the argument:
sn(12K, k) = √ 1
1+ k , cn(
1
2K, k) =
k
1+ k , dn(
1
2K, k) =
√
k ;
sn(K, k) =1, cn(K, k) =0, dn(K, k) = k
18.14.1-2 Reduction formulas
sn(u K) = cn u
dn u, cn(u K) =k sn u
k
dn u; sn(u 2K) = – sn u, cn(u 2K) = – cn u, dn(u 2K) = dn u;
sn(u + iK) = 1
k sn u, cn(u + iK
) = –i
k
dn u
sn u, dn(u + iK
) = –i cn u
sn u; sn(u +2iK ) = sn u, cn(u +2iK ) = – cn u, dn(u +2iK ) = – dn u;
sn(u +K+iK) = dn u
k cn u, cn(u +K+iK
) = –i k
k cn u, dn(u +K+iK
) = ik sn u
cn u; sn(u +2K+2iK ) = – sn u, cn(u +2K+2iK ) = cn u, dn(u +2K+2iK ) = – dn u.
18.14.1-3 Periods, zeros, poses, and residues
TABLE 18.4 Periods, zeros, poles, and residues of the Jacobian elliptic functions
( m, n = 0 , 1 , 2, ; i2 = – 1 )
sn u 4mK + 2nK i 2mK + 2nK i 2mK +( 2n+ 1 ) K i (– 1 )m k1
cn u ( 4m+ 2n) K + 2nK i ( 2m+ 1 ) K + 2nK i 2mK +( 2n+ 1 ) K i (– 1 )m–1k
dn u 2mK + 4nK i ( 2m+ 1 ) K +( 2n+ 1 ) K i 2mK +( 2n+ 1 ) K i (– 1 )n–1i
Trang 6974 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
18.14.1-4 Double-argument formulas
sn(2u) = 2sn u cn u dn u
1– k2sn4u = 2sn u cn u dn u
cn2u+ sn2udn2u, cn(2u) = cn
2u– sn2udn2u
1– k2sn4u = cn
2u– sn2udn2u
cn2u+ sn2udn2u, dn(2u) = dn
2u – k2sn2ucn2u
1– k2sn4u = dn
2u+ cn2u(dn2u–1)
dn2u– cn2u(dn2u–1).
18.14.1-5 Half-argument formulas
sn2 u
2 =
1
k2
1– dn u
1+ cn u =
1– cn u
1+ dn u,
cn2 u
2 =
cn u + dn u
1+ dn u =
1– k2
k2
1– dn u
dn u – cn u,
dn2 u
2 =
cn u + dn u
1+ cn u = (1– k2) 1– cn u
dn u – cn u.
18.14.1-6 Argument addition formulas
sn(u v) = sn u cn v dn v sn v cn u dn u
1– k2sn2usn2v ,
cn(u v) = cn u cn vsn u sn v dn u dn v
1– k2sn2usn2v ,
dn(u v) = dn u dn vk2sn u sn v cn u cn v
1– k2sn2usn2v
18.14.1-7 Conversion formulas
Table 18.5 presents conversion formulas for Jacobi elliptic functions If k > 1, then
k1 =1/k < 1 Elliptic functions with real modulus can be reduced, using the first set of conversion formulas, to elliptic functions with a modulus lying between 0 and 1
18.14.1-8 Descending Landen transformation (Gauss’s transformation)
Notation:
μ=
11– k + k
, v = 1+ μ u Descending transformations:
sn(u, k) = (1+ μ) sn(v, μ2)
1+ μ sn2(v, μ2), cn(u, k) =
cn(v, μ2) dn(v, μ2)
1+ μ sn2(v, μ2) , dn(u, k) =
dn2(v, μ2) + μ –1
1+ μ – dn2(v, μ2).
Trang 7TABLE 18.5
Conversion formulas for Jacobi elliptic functions Full notation is used: sn(u, k), cn(u, k), dn(u, k)
cn(u, k)
1
cn(u, k)
dn(u, k) cn(u, k)
sn(u, k) dn(u, k)
cn(u, k) dn(u, k)
1
dn(u, k)
sn(u, k) dn(u, k)
1
dn(u, k)
cn(u, k) dn(u, k)
sn(u, k) cn(u, k)
dn(u, k) cn(u, k)
1
cn(u, k)
( 1+ k)u 2√ k
1+ k
( 1+ k) sn(u, k)
1+ k sn2(u, k)
cn(u, k) dn(u, k)
1+ k sn2(u, k)
1– k sn2(u, k)
1+ k sn2(u, k)
( 1+ k )u 1– k
1+ k
( 1+ k ) sn(u, k) cn(u, k) dn(u, k)
1 – ( 1+ k ) sn2(u, k) dn(u, k)
1 – ( 1– k ) sn2(u, k) dn(u, k)
18.14.1-9 Ascending Landen transformation
Notation:
μ= 4k
(1+ k)2, σ=
11– k
+ k
, v= u
1+ σ.
Ascending transformations:
sn(u, k) = (1+ σ) sn(v, μ) cn(v, μ)
dn(v, μ) , cn(u, k) =
1+ σ μ
dn2(v, μ) – σ dn(v, μ) , dn(u, k) =
1– σ μ
dn2(v, μ) + σ dn(v, μ) .
18.14.1-10 Series representation
Representation Jacobi functions in the form of power series in u:
sn u = u – 1
3!(1+ k2)u3+ 1
5!(1+14k2+ k4)u5– 1
7!(1+135k2+135k4+ k6)u7+· · · ,
cn u =1– 1
2!u
2+ 1
4!(1+4k2)u4– 1
6!(1+44k2+16k4)u6+· · · ,
dn u =1– 1
2!k
2u2+ 1
4!k
2(4+ k2)u4– 1
6!k
2(16+44k2+ k4)u6+· · · ,
am u = u – 1
3!k
2u3+ 1
5!k
2(4+ k2)u5– 1
7!k
2(16+44k2+ k4)u7+· · ·
These functions converge for|u|<|K(k )|
Representation Jacobi functions in the form of trigonometric series:
sn u = 2π
kK√ q
∞
n=1
q n
1– q2n–1 sin
(2n–1)πu
2K
,
...18.13.1-2 Representation of complete elliptic integrals in series form
Representation of complete elliptic integrals in the form of series in powers of the modulus k:
K(k)... reduced, using the first set of conversion formulas, to elliptic functions with a modulus lying between and
18.14.1-8 Descending Landen transformation (Gauss’s transformation)
Notation:... data-page="2">
970 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
Representation of complete elliptic integrals in the form of series in powers of the
comple-mentary modulus k