Series Representation of the Jacobi Theta Functions.. Definition of the Jacobi theta functions.. The theta functions are not elliptic functions.. The very good convergence of their serie
Trang 1976 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
cn u = 2π
kK√ q
∞
n=1
q n
1+ q2n–1 cos
(2n–1)πu
2K
,
dn u = π
2K + 2π
K
∞
n=1
q n
1+ q2n cos
nπu
K
,
am u = πu
2K +2∞
n=1
1
n
q n
1+ q2n sin
nπu
K
,
where q = exp(–πK /K),K=K(k),K =K(k ), and k =√
1– k2
18.14.1-11 Derivatives and integrals
Derivatives:
d
du sn u = cn u dn u, d
du cn u = – sn u dn u, d
du dn u = –k2sn u cn u.
sn u du = 1
k ln(dn u – k cn u) = –1
k ln(dn u + k cn u),
cn u du = 1
k arccos(dn u) = 1
k arcsin(k sn u),
dn u du = arcsin(sn u) = am u.
The arbitrary additive constant C in the integrals is omitted.
18.14.2 Weierstrass Elliptic Function
18.14.2-1 Infinite series representation Some properties
The Weierstrass elliptic function (or Weierstrass ℘-function) is defined as
℘ (z) = ℘(z|ω1, ω2) = 1
z2 +
m,n
(z –2mω1–2nω2)2 –
1
(2mω1+2nω2)2
,
where the summation is assumed over all integer m and n, except for m = n =0 This
function is a complex, double periodic function of a complex variable z with periods2ω1
and2ω1:
℘ (–z) = ℘(z),
℘ (z +2mω1+2nω2) = ℘(z),
where m, n = 0, 1, 2, and Im(ω2/ω1) ≠ 0 The series defining the Weierstrass ℘-function converges everywhere except for second-order poles located at z mn=2mω1+2nω2.
Argument addition formula:
℘ (z1+ z2) = –℘(z1) – ℘(z2) + 1
4
℘ (z1) – ℘ (z2)
℘ (z1) – ℘(z2)
2
Trang 218.14.2-2 Representation in the form of a definite integral.
The Weierstrass function ℘ = ℘(z, g2, g3) = ℘(z|ω1, ω2) is defined implicitly by the elliptic integral:
z=
℘
dt
4t3– g2t – g3 =
∞
℘
dt
2√ (t – e1)(t – e2)(t – e3).
The parameters g2and g3are known as the invariants.
The parameters e1, e2, e3, which are the roots of the cubic equation4z3– g2z – g3=0,
are related to the half-periods ω1, ω2and invariants g2, g3by
e1= ℘(ω1), e2= ℘(ω1+ ω2), e1= ℘(ω2),
e1+ e2+ e3 =0, e1e2+ e1e3+ e2e3= –14g2, e1e2e3= 14g3.
Homogeneity property:
℘ (z, g2, g3) = λ2℘ (λz, λ–4g2, λ–6g3).
18.14.2-3 Representation as a Laurent series Differential equations
The Weierstrass ℘-function can be expanded into a Laurent series:
℘ (z) = 1
z2 +
g2
20z2+
g3
28z4+
g2 2
1200z6+
3g2g3
6160 z8+· · · =
1
z2 +
∞
k=2
a k z2k–2,
(k –3)(2k+1)
k–2
m=2
a m a k–m for k≥ 4, 0<|z|< min(|ω1|,|ω2|)
The Weierstrass ℘-function satisfies the first-order and second-order nonlinear
differen-tial equations:
(℘ z)2=4℘3– g
2℘ – g3,
℘
zz =6℘2– 1
2g2.
18.14.2-4 Connection with Jacobi elliptic functions
Direct and inverse representations of the Weierstrass elliptic function via Jacobi elliptic functions:
℘ (z) = e1+ (e1– e3)cn
2w
sn2w = e2+ (e1– e3)dn
2w
sn2w = e3+ e1– e3
sn2w ;
sn w = e1– e3
℘ (z) – e3, cn w =
!
℘ (z) – e1
℘ (z) – e3, dn w =
!
℘ (z) – e2
℘ (z) – e3;
w = z √
e1– e3=Kz/ω1.
The parameters are related by
k= e2– e3
e1– e3, k
= e1– e2
e1– e3, K= ω1
√
e1– e3, iK = ω
2√
e1– e3.
Trang 3978 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
18.15 Jacobi Theta Functions
18.15.1 Series Representation of the Jacobi Theta Functions.
Simplest Properties
18.15.1-1 Definition of the Jacobi theta functions
The Jacobi theta functions are defined by the following series:
ϑ1(v) = ϑ1(v, q) = ϑ1(v|τ) = 2∞
n=0
(– 1 )n q(n+1/2 )2sin[( 2n+ 1)πv] = i
∞
n=–∞
(– 1 )n q(n–1/2 )2e iπ(2n–1 )v,
ϑ2(v) = ϑ2(v, q) = ϑ2(v|τ) = 2∞
n=0
q(n+1/2 )2cos[( 2n+ 1)πv] =
∞
n=–∞
q(n–1/2 )2e iπ(2n–1 )v,
ϑ3(v) = ϑ3(v, q) = ϑ3(v|τ) = 1 + 2∞
n=0
q n2cos( 2nπv) =
∞
n=–∞
q n2e2iπnv,
ϑ4(v) = ϑ4(v, q) = ϑ4(v|τ) = 1 + 2∞
n=0
(– 1 )n q n2
cos( 2nπv) =
∞
n=–∞
(– 1 )n q n2
e2iπnv,
where v is a complex variable and q = e iπτ is a complex parameter (τ has a positive
imaginary part)
18.15.1-2 Simplest properties
The Jacobi theta functions are periodic entire functions that possess the following properties:
ϑ1(v) odd, has period 2, vanishes at v = m + nτ ;
ϑ2(v) even, has period 2, vanishes at v = m + nτ + 12;
ϑ3(v) even, has period 1, vanishes at v = m + (n + 12)τ + 12;
ϑ4(v) even, has period 1, vanishes at v = m + (n + 12)τ
Here, m, n =0, 1, 2,
Remark The theta functions are not elliptic functions The very good convergence of their series allows the computation of various elliptic integrals and elliptic functions using the relations given above in Paragraph 18.15.1-1.
18.15.2 Various Relations and Formulas Connection with Jacobi
Elliptic Functions
18.15.2-1 Linear and quadratic relations
Linear relations (first set):
ϑ1
v+ 1 2
= ϑ2(v), ϑ2
v+ 1 2
= –ϑ1(v),
ϑ3
v+ 1 2
= ϑ4(v), ϑ4
v+ 1 2
= ϑ3(v),
ϑ1
v+ τ
2
= ie–iπ v+ τ4
ϑ4(v), ϑ2
v+ τ
2
= e–iπ v+ τ4
ϑ3(v),
ϑ3
v+ τ
2
= e–iπ v+ τ4
ϑ2(v), ϑ4
v+ τ
2
= ie–iπ v+ τ4
ϑ1(v).
Trang 4Linear relations (second set):
ϑ1(v|τ+1) = e iπ/4ϑ1(v|τ), ϑ2(v|τ +1) = e iπ/4ϑ2(v|τ),
ϑ3(v|τ +1) = ϑ4(v|τ), ϑ4(v|τ +1) = ϑ3(v|τ),
ϑ1
v
τ
–1
τ
= 1
i
τ
i e
iπv2/τ ϑ1(v|τ), ϑ2
v
τ
–1
τ
i e
iπv2/τ ϑ4(v|τ),
ϑ3
v
τ
–1
τ
i e
iπv2/τ ϑ3(v|τ), ϑ4
v
τ
–1
τ
i e
iπv2/τ ϑ2(v|τ)
Quadratic relations:
ϑ2
1(v)ϑ22(0) = ϑ24(v)ϑ23(0) – ϑ23(v)ϑ24(0),
ϑ2
1(v)ϑ23(0) = ϑ24(v)ϑ22(0) – ϑ22(v)ϑ24(0),
ϑ2
1(v)ϑ24(0) = ϑ23(v)ϑ22(0) – ϑ22(v)ϑ23(0),
ϑ2
4(v)ϑ24(0) = ϑ23(v)ϑ23(0) – ϑ22(v)ϑ22(0)
18.15.2-2 Representation of the theta functions in the form of infinite products
ϑ1(v) =2q0q1 4sin(πv)∞
n=1
1–2q2ncos(2πv ) + q4n
,
ϑ2(v) =2q0q1 4cos(πv)∞
n=1
1+2q2ncos(2πv ) + q4n
,
ϑ3(v) = q0
∞
n=1
1+2q2n–1cos(2πv ) + q4n–2
,
ϑ4(v) = q0
∞
n=1
1–2q2n–1cos(2πv ) + q4n–2
,
where q0= ∞
n=1(1– q2n)
18.15.2-3 Connection with Jacobi elliptic functions
Representations of Jacobi elliptic functions in terms of the theta functions:
sn w = ϑ3(0)
ϑ2(0)
ϑ1(v)
ϑ4(v), cn w =
ϑ4(0)
ϑ2(0)
ϑ2(v)
ϑ4(v), dn w =
ϑ4(0)
ϑ3(0)
ϑ3(v)
ϑ4(v), w=2Kv The parameters are related by
k= ϑ
2
2(0)
ϑ2
3(0), k
= ϑ24(0)
ϑ2
3(0), K=
π
2ϑ23(0), K = –iτK.
Trang 5980 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
TABLE 18.6 The Mathieu functions cen= cen (x, q) and se n= sen (x, q) (for odd n, functions
cenand senare 2π -periodic, and for even n, they are π-periodic); definite
eigenvalues a = a n (q) and a = b n (q) correspond to each value of parameter q
Mathieu functions Recurrence relationsfor coefficients Normalizationconditions
ce2n=
∞
m=0
A22n mcos 2mx
qA22n = a2n A20n;
qA24n = (a2n– 4)A22n– 2qA20n;
qA22n m+2= (a2n– 4m2)A22n m –qA22n m–2, m≥ 2
(A20n)2+
∞
m=0
(A22n m)2
=
2 if n =0
1 if n≥ 1
ce 2n+1 =
∞
m=0
A22n+ m+11cos( 2m+ 1)x
qA23n+1= (a2n+1 – 1–q)A21n+1;
qA22n+ m+13= [a2n+1–( 2m+ 1 )2]A22n+ m+11 –qA22n+ m–11, m≥ 1
∞
m=0
(A22n+ m+11)2= 1
se2n=
∞
m=0
B22n msin 2mx,
se 0 = 0
qB42n = (b2n– 4)B22n;
qB22n m+2= (b2n– 4m2)B22m n –qB22m– n 2, m≥ 2
∞
m=0
(B22m n)2= 1
se 2n+1 =
∞
m=0
B22m+ n+11sin( 2m+ 1)x
qB32n+1= (b2n+1 – 1–q)B12n+1;
qB22n+ m+13= [b2n+1–( 2m+ 1 )2]B22n+ m+11 –qB22m– n+11, m≥ 1
∞
m=0
(B22m+ n+11)2= 1
18.16 Mathieu Functions and Modified Mathieu
Functions
18.16.1 Mathieu Functions
18.16.1-1 Mathieu equation and Mathieu functions
The Mathieu functions cen (x, q) and se n (x, q) are periodical solutions of the Mathieu
equation
y
xx + (a –2qcos2x )y =0
Such solutions exist for definite values of parameters a and q (those values of a are referred
to as eigenvalues) The Mathieu functions are listed in Table 18.6
18.16.1-2 Properties of the Mathieu functions
The Mathieu functions possess the following properties:
ce2n (x, –q) = (–1)nce2nπ
2– x, q
, ce2n+1(x, –q) = (–1)nse2n+1π
2 – x, q
,
se2n (x, –q) = (–1)n–1se2n
π
2– x, q
, se2n+1(x, –q) = (–1)nce2n+1
π
2 – x, q
Selecting sufficiently large number m and omitting the term with the maximum number
in the recurrence relations (indicated in Table 18.6), we can obtain approximate relations
for eigenvalues a n (or b n ) with respect to parameter q Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients A n m (or B m n) to
zero, we obtain an algebraic equation for finding a n (q) (or b n (q)).
Trang 6For fixed real q≠ 0, eigenvalues a n and b nare all real and different, while
if q >0 then a0 < b1 < a1< b2 < a2<· · · ;
if q <0 then a0 < a1< b1< b2 < a2< a3 < b3< b4<· · ·
The eigenvalues possess the properties
a2n (–q) = a2n (q), b2n (–q) = b2n (q), a2n+1(–q) = b2n+1(q).
Tables of the eigenvalues a n = a n (q) and b n = b n (q) can be found in Abramowitz and
Stegun (1964, chap 20)
The solution of the Mathieu equation corresponding to eigenvalue a n (or b n ) has n zeros
on the interval0 ≤x < π (q is a real number).
18.16.1-3 Asymptotic expansions as q →0and q → ∞.
Listed below are two leading terms of asymptotic expansions of the Mathieu functions
cen (x, q) and se n (x, q), as well as of the corresponding eigenvalues a n (q) and b n (q), as
q →0:
ce0(x, q) = 1
√
2
1– q
2 cos2x
, a0(q) = – q
2
2 +
7q4
128;
ce1(x, q) = cos x – q
8cos3x, a1(q) =1+ q;
ce2(x, q) = cos2x+ q
4
1– cos4x
3
, a2(q) =4+ 5q2
12 ;
cen (x, q) = cos nx + q
4
cos(n +2)x
cos(n –2)x
n–1
, a n (q) = n2+ q
2
2(n2–1) (n≥ 3);
se1(x, q) = sin x – q
8 sin3x, b1(q) =1– q;
se2(x, q) = sin2x – qsin4x
12 , b2(q) =4–
q2
12;
sen (x, q) = sin nx – q
4
sin(n +2)x
sin(n –2)x
n–1
, b n (q) = n2+ q
2
2(n2–1) (n≥ 3)
Asymptotic results as q → ∞ (–π/2< x < π/2):
a n (q)≈–2q+2(2n+1)√
q+ 14(2n2+2n+1),
b n+1(q)≈–2q+2(2n+1)√
q+ 14(2n2+2n+1),
cen (x, q)≈λ n q–1 4cos–n–1x
cos2n+1ξexp(2√ q sin x) + sin2n+1ξexp(–2√ q sin x)
,
sen+1(x, q)≈μ n+1q–1 4cos–n–1x
cos2n+1ξexp(2√ q sin x) – sin2n+1ξexp(–2√ q sin x)
,
where λ n and μ n are some constants independent of the parameter q, and ξ = 12x+ π4
Trang 7982 SPECIALFUNCTIONS ANDTHEIRPROPERTIES
18.16.2 Modified Mathieu Functions
The modified Mathieu functions Cen (x, q) and Se n (x, q) are solutions of the modified
Mathieu equation
y
xx – (a –2qcosh2x )y =0,
with a = a n (q) and a = b n (q) being the eigenvalues of the Mathieu equation (see Subsection
18.16.1)
The modified Mathieu functions are defined as
Ce2n+p (x, q) = ce2n+p (ix, q) =
∞
k=0
A2n+p
2k+pcosh[(2k + p)x],
Se2n+p (x, q) = –i se2n+p (ix, q) =
∞
k=0
B2n+p
2k+psinh[(2k + p)x],
where p may be equal to 0 and 1, and coefficients A22n+p k+p and B22n+p k+p are indicated in Subsection 18.16.1
18.17 Orthogonal Polynomials
All zeros of each of the orthogonal polynomialsP n (x) considered in this section are real
and simple The zeros of the polynomialsP n (x) and P n+1(x) are alternating.
For Legendre polynomials see Subsection 18.11.1
18.17.1 Laguerre Polynomials and Generalized Laguerre
Polynomials
18.17.1-1 Laguerre polynomials
The Laguerre polynomials L n = L n (x) satisfy the second-order linear ordinary differential
equation
xy
xx+ (1– x)y x + ny =0
and are defined by the formulas
L n (x) = 1
n!e
x d n
dx n x n e–x
= (–1)n
n!
x n – n2x n–1+ n2(n –1)2
2! x
n–2+· · ·
The first four polynomials have the form
L0(x) =1, L1(x) = –x +1, L2(x) = 12(x2–4x+2), L3(x) = 16(–x3+9x2–18x+6)
To calculate L n (x) for n≥ 2, one can use the recurrence formulas
L n+1(x) = 1
n+1
(2n+1– x)L n (x) – nL n–1(x)
The functions L n (x) form an orthonormal system on the interval 0 < x < ∞ with
0 e
–x L
n (x)L m (x) dx =
0
if n≠m,
1 if n = m.
The generating function is
1
1– sexp
– sx
1– s
=
∞
n=0
L n (x)s n, |s|<1
... exist for definite values of parameters a and q (those values of a are referredto as eigenvalues) The Mathieu functions are listed in Table 18.6
18.16.1-2 Properties of the... →0and q → ∞.
Listed below are two leading terms of asymptotic expansions of the Mathieu functions
cen (x, q) and se n (x, q), as well as of the...
All zeros of each of the orthogonal polynomialsP n (x) considered in this section are real
and simple The zeros of the polynomialsP n (x) and P n+1(x)