18.13. ELLIPTIC INTEGRALS 969
18.13. 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 – k
2
sin
2
α
=
1
0
dx
(1 – x
2
)(1 – k
2
x
2
)
.
Complete elliptic integral of the second kind:
E(k)=
π/2
0
√
1 – k
2
sin
2
αdα=
1
0
√
1 – k
2
x
2
√
1 – x
2
dx.
The argument k is called the elliptic modulus (k
2
< 1).
Notation:
k
=
√
1 – k
2
, 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
1 + k
E(k)+k
K(k)
,
K
2
√
k
1 + k
=(1 + k) K(k),
E
2
√
k
1 + k
=
1
1 + k
2 E(k)–(k
)
2
K(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
k
2
+
1×3
2×4
2
k
4
+ ···+
(2n – 1)!!
(2n)!!
2
k
2n
+ ···
,
E(k)=
π
2
1 –
1
2
2
k
2
1
–
1×3
2×4
2
k
4
3
– ···–
(2n – 1)!!
(2n)!!
2
k
2n
2n – 1
– ···
.
970 SPECIAL FUNCTIONS AND THEIR PROPERTIES
Representation of complete elliptic integrals in the form of series in powers of the comple-
mentary modulus k
=
√
1 – k
2
:
K(k)=
π
1 +k
1 +
1
2
2
1 –k
1 +k
2
+
1×3
2×4
2
1 –k
1 +k
4
+ ···+
(2n –1)!!
(2n)!!
2
1 –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)!!
2
1 –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:
d K(k)
dk
=
E(k)
k(k
)
2
–
K(k)
k
,
d E(k)
dk
=
E(k)–K(k)
k
.
The functions K(k)andK
(k) satisfy the second-order linear ordinary differential equa-
tion
d
dk
k(1 – k
2
)
d K
dk
– k K = 0.
The functions E(k)andE
(k)–K
(k) satisfy the second-order linear ordinary differential
equation
(1 – k
2
)
d
dk
k
d E
dk
+ k E = 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 – k
2
sin
2
α
=
sin ϕ
0
dx
(1 – x
2
)(1 – k
2
x
2
)
.
Elliptic integral of the second kind:
E(ϕ, k)=
ϕ
0
√
1 – k
2
sin
2
αdα=
sin ϕ
0
√
1 – k
2
x
2
√
1 – x
2
dx.
18.13. ELLIPTIC INTEGRALS 971
Elliptic integral of the third kind:
Π(ϕ, n, k)=
ϕ
0
dα
(1 – n sin
2
α)
√
1 – k
2
sin
2
α
=
sin ϕ
0
dx
(1 – nx
2
)
(1 – x
2
)(1 – k
2
x
2
)
.
The quantity k is called the elliptic modulus (k
2
< 1), k
=
√
1 – k
2
is the complementary
modulus,andn is the characteristic parameter.
Complete elliptic integrals:
K(k)=F
π
2
, k
, E(k)=E
π
2
, k
,
K
(k)=F
π
2
, k
, E
(k)=E
π
2
, k
.
Properties of elliptic integrals:
F (–ϕ, k)=–F (ϕ, k), F (nπ
ϕ, k)=2n K(k) F (ϕ, k);
E(–ϕ, k)=–E(ϕ, k), E(nπ
ϕ, k)=2n E(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
)
2
F (ϕ, k)
,
where the angles ϕ and ψ are related by sin ψ = k sin ϕ,cosψ =
1 – k
2
sin
2
ϕ.
Conversion formulas for elliptic integrals (second set):
F
ψ,
1 – k
1 + k
=(1 + k
)F (ϕ, k),
E
ψ,
1 – k
1 + k
=
2
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
1 + k
2E(ϕ, k)–(k
)
2
F (ϕ, k)+2k
sin ϕ cos ϕ
1 + k sin
2
ϕ
1 – k
2
sin
2
ϕ
,
where the angles ϕ and ψ are related by sin ψ =
(1 + k)sinϕ
1 + k sin
2
ϕ
.
972 SPECIAL FUNCTIONS AND THEIR PROPERTIES
18.13.2-3. Trigonometric expansions.
Trigonometric expansions for small k and ϕ:
F (ϕ, k)=
2
π
K(k)ϕ –sinϕ cos ϕ
a
0
+
2
3
a
1
sin
2
ϕ +
2×4
3×5
a
2
sin
4
ϕ + ···
,
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)ϕ –sinϕ cos ϕ
b
0
+
2
3
b
1
sin
2
ϕ +
2×4
3×5
b
2
sin
4
ϕ + ···
,
b
0
= 1 –
2
π
E(k), b
n
= b
n–1
–
(2n – 1)!!
(2n)!!
2
k
2n
2n – 1
.
Trigonometric expansions for k → 1:
F (ϕ, k)=
2
π
K
(k)lntan
ϕ
2
+
π
4
–
tan ϕ
cos ϕ
a
0
–
2
3
a
1
tan
2
ϕ +
2×4
3×5
a
2
tan
4
ϕ – ···
,
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)lntan
ϕ
2
+
π
4
+
tan ϕ
cos ϕ
b
0
–
2
3
b
1
tan
2
ϕ +
2×4
3×5
b
2
tan
4
ϕ – ···
,
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 2mω
1
+ 2nω
2
with integer m and n,whereω
1
and ω
2
are a pair of
(primitive) half-periods. The ratio τ = ω
2
/ω
1
is 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 – k
2
.
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 – k
2
sin
2
α
= F (ϕ, k)
is treated as a function of u, the following notation is used:
u =amϕ.
18.14. ELLIPTIC FUNCTIONS 973
Naming: ϕ is the amplitude and u is the argument.
Jacobi elliptic functions:
sn u =sinϕ =sinamu (sine amplitude),
cn u =cosϕ =cosamu (cosine amplitude),
dn u =
1 – k
2
sin
2
ϕ =
dϕ
du
(delta amlplitude).
Along with the brief notations sn u,cnu,dnu, the respective full notations are also used:
sn(u, k), cn(u, k), dn(u, k).
Simple properties:
sn(–u)=–snu,cn(–u)=cnu,dn(–u)=dnu;
sn
2
u +cn
2
u = 1, k
2
sn
2
u +dn
2
u = 1,dn
2
u – k
2
cn
2
u = 1 – k
2
,
where i
2
=–1.
Jacobi functions for special values of the modulus (k = 0 and k = 1):
sn(u, 0)=sinu,cn(u, 0)=cosu,dn(u, 0)=1;
sn(u, 1)=tanhu,cn(u, 1)=
1
cosh u
,dn(u, 1)=
1
cosh u
.
Jacobi functions for special values of the argument:
sn(
1
2
K, k)=
1
√
1 + k
,cn(
1
2
K, k)=
k
1 + k
,dn(
1
2
K, 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
dn u
, dn(u K)=
k
dn u
;
sn(u 2 K)=–snu,cn(u 2 K)=–cnu, dn(u 2 K)=dnu;
sn(u + i K
)=
1
k sn u
,cn(u + i K
)=–
i
k
dn u
sn u
, dn(u + i K
)=–i
cn u
sn u
;
sn(u + 2i K
)=snu,cn(u + 2i K
)=–cnu, dn(u + 2i K
)=–dnu;
sn(u + K +i K
)=
dn u
k cn u
,cn(u + K+i K
)=–i
k
k cn u
, dn(u + K+i K
)=ik
sn u
cn u
;
sn(u + 2 K +2i K
)=–snu,cn(u + 2 K +2i K
)=cnu, dn(u + 2 K +2i K
)=–dnu.
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, ; i
2
=–1)
Functions Periods Zeros Poles Residues
sn u
4m K +2n K
i 2m K +2n K
i
2m K +(2n +1) K
i
(–1)
m
1
k
cn u
(4m+2n) K+2n K
i (2m+1) K+2n K
i 2m K +(2n +1) K
i
(–1)
m–1
i
k
dn u
2m K +4n K
i
(2m+1) K+(2n+1) K
i 2m K +(2n +1) K
i
(–1)
n–1
i
974 SPECIAL FUNCTIONS AND THEIR PROPERTIES
18.14.1-4. Double-argument formulas.
sn(2u)=
2 sn u cn u dn u
1 – k
2
sn
4
u
=
2 sn u cn u dn u
cn
2
u +sn
2
u dn
2
u
,
cn(2u)=
cn
2
u –sn
2
u dn
2
u
1 – k
2
sn
4
u
=
cn
2
u –sn
2
u dn
2
u
cn
2
u +sn
2
u dn
2
u
,
dn(2u)=
dn
2
u – k
2
sn
2
u cn
2
u
1 – k
2
sn
4
u
=
dn
2
u +cn
2
u (dn
2
u – 1)
dn
2
u –cn
2
u (dn
2
u – 1)
.
18.14.1-5. Half-argument formulas.
sn
2
u
2
=
1
k
2
1 –dnu
1 +cnu
=
1 –cnu
1 +dnu
,
cn
2
u
2
=
cn u +dnu
1 +dnu
=
1 – k
2
k
2
1 –dnu
dn u –cnu
,
dn
2
u
2
=
cn u +dnu
1 +cnu
=(1 – k
2
)
1 –cnu
dn u –cnu
.
18.14.1-6. Argument addition formulas.
sn(u v)=
sn u cn v dn v sn v cn u dn u
1 – k
2
sn
2
u sn
2
v
,
cn(u
v)=
cn u cn v sn u sn v dn u dn v
1 – k
2
sn
2
u sn
2
v
,
dn(u
v)=
dn u dn v k
2
sn u sn v cn u cn v
1 – k
2
sn
2
u sn
2
v
.
18.14.1-7. Conversion formulas.
Table 18.5 presents conversion formulas for Jacobi elliptic functions. If k > 1,then
k
1
= 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 =
1 – k
1 + k
, v =
u
1 + μ
.
Descending transformations:
sn(u, k)=
(1 + μ)sn(v, μ
2
)
1 + μ sn
2
(v, μ
2
)
,cn(u, k)=
cn(v, μ
2
)dn(v, μ
2
)
1 + μ sn
2
(v, μ
2
)
,dn(u, k)=
dn
2
(v, μ
2
)+μ – 1
1 + μ –dn
2
(v, μ
2
)
.
18.14. ELLIPTIC FUNCTIONS 975
TABLE 18.5
Conversion formulas for Jacobi elliptic functions. Full notation is used: sn(u, k), cn(u, k), dn(u, k)
u
1
k
1
sn(u
1
, k
1
) cn(u
1
, k
1
) dn(u
1
, k
1
)
ku
1
k
k sn(u, k) dn(u, k) cn(u, k)
iu
k
i
sn(u, k)
cn(u, k)
1
cn(u, k)
dn(u, k)
cn(u, k)
k
u
i
k
k
k
sn(u, k)
dn(u, k)
cn(u, k)
dn(u, k)
1
dn(u, k)
iku
i
k
k
ik
sn(u, k)
dn(u, k)
1
dn(u, k)
cn(u, k)
dn(u, k)
ik
u
1
k
ik
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 sn
2
(u, k)
cn(u, k)dn(u, k)
1 +k sn
2
(u, k)
1 –k sn
2
(u, k)
1 +k sn
2
(u, k)
(1 +k
)u
1 –k
1 +k
(1 +k
)sn(u, k)cn(u, k)
dn(u, k)
1 –(1 +k
)sn
2
(u, k)
dn(u, k)
1 –(1 –k
)sn
2
(u, k)
dn(u, k)
18.14.1-9. Ascending Landen transformation.
Notation:
μ =
4k
(1 + k)
2
, σ =
1 – k
1 + 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
dn
2
(v, μ)–σ
dn(v, μ)
,dn(u, k)=
1 –σ
μ
dn
2
(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 + k
2
)u
3
+
1
5!
(1 + 14k
2
+ k
4
)u
5
–
1
7!
(1 + 135k
2
+ 135k
4
+ k
6
)u
7
+ ··· ,
cn u = 1 –
1
2!
u
2
+
1
4!
(1 + 4k
2
)u
4
–
1
6!
(1 + 44k
2
+ 16k
4
)u
6
+ ··· ,
dn u = 1 –
1
2!
k
2
u
2
+
1
4!
k
2
(4 + k
2
)u
4
–
1
6!
k
2
(16 + 44k
2
+ k
4
)u
6
+ ··· ,
am u = u –
1
3!
k
2
u
3
+
1
5!
k
2
(4 + k
2
)u
5
–
1
7!
k
2
(16 + 44k
2
+ k
4
)u
7
+ ··· .
These functions converge for |u| < |K(k
)|.
Representation Jacobi functions in the form of trigonometric series:
sn u =
2π
k K
√
q
∞
n=1
q
n
1 – q
2n–1
sin
(2n – 1)
πu
2 K
,