Handbook of mathematics for engineers and scienteists part 145 ppsx

976 SPECIAL FUNCTIONS AND THEIR PROPERTIES
cn u =
2π
k K
√
q
∞

n=1
q
n
1 + q
2n–1
cos

(2n – 1)
πu
2 K

,
dn u =
π
2 K
+
2π
K
∞

n=1
q
n

1 + q
2n
cos

nπu
K

,
am u =
πu
2 K
+ 2
∞

n=1
1
n
q
n
1 + q
2n
sin

nπu
K

,
where q =exp(–π K

/ K), K = K(k), K


= K(k

), and k

=
√
1 – k
2
.
18.14.1-11. Derivatives and integrals.
Derivatives:
d
du
sn u =cnu dn u,
d
du
cn u =–snu dn u,
d
du
dn u =–k
2
sn u cn u.
Integrals:

sn udu=
1
k
ln(dn u – k cn u)=–
1

k
ln(dn u + k cn u),

cn udu=
1
k
arccos(dn u)=
1
k
arcsin(k sn u),

dn udu= arcsin(sn u)=amu.
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|ω
1
, ω
2
)=
1
z
2
+

m,n

1
(z – 2mω

1
– 2nω
2
)
2
–
1
(2mω
1
+ 2nω
2
)
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 periods 2ω
1
and 2ω
1
:
℘(–z)=℘(z),
℘(z + 2mω
1
+ 2nω
2
)=℘(z),
where m, n = 0,
1, 2, and Im(ω
2

/ω
1
) ≠ 0. The series defining the Weierstrass ℘-
function converges everywhere except for second-order poles located at z
mn
=2mω
1
+2nω
2
.
Argument addition formula:
℘(z
1
+ z
2
)=–℘(z
1
)–℘(z
2
)+
1
4

℘

(z
1
)–℘

(z

2
)
℘(z
1
)–℘(z
2
)

2
.
18.14. ELLIPTIC FUNCTIONS 977
18.14.2-2. Representation in the form of a definite integral.
The Weierstrass function ℘ = ℘(z, g
2
, g
3
)=℘(z|ω
1
, ω
2
)isdefined implicitly by the elliptic
integral:
z =

∞
℘
dt

4t
3

– g
2
t – g
3
=

∞
℘
dt
2
√
(t – e
1
)(t – e
2
)(t – e
3
)
.
The parameters g
2
and g
3
are known as the invariants.
The parameters e
1
, e
2
, e
3

, which are the roots of the cubic equation 4z
3
– g
2
z – g
3
= 0,
are related to the half-periods ω
1
, ω
2
and invariants g
2
, g
3
by
e
1
= ℘(ω
1
), e
2
= ℘(ω
1
+ ω
2
), e
1
= ℘(ω
2

),
e
1
+ e
2
+ e
3
= 0, e
1
e
2
+ e
1
e
3
+ e
2
e
3
=–
1
4
g
2
, e
1
e
2
e
3

=
1
4
g
3
.
Homogeneity property:
℘(z, g
2
, g
3
)=λ
2
℘(λz, λ
–4
g
2
, λ
–6
g
3
).
18.14.2-3. Representation as a Laurent series. Differential equations.
The Weierstrass ℘-function can be expanded into a Laurent series:
℘(z)=
1
z
2
+
g

2
20
z
2
+
g
3
28
z
4
+
g
2
2
1200
z
6
+
3g
2
g
3
6160
z
8
+ ··· =
1
z
2
+

∞

k=2
a
k
z
2k–2
,
a
k
=
3
(k – 3)(2k + 1)
k–2

m=2
a
m
a
k–m
for k ≥ 4, 0 < |z| <min(|ω
1
|, |ω
2
|).
The Weierstrass ℘-function satisfies the first-order and second-order nonlinear differen-
tial equations:
(℘

z

)
2
= 4℘
3
– g
2
℘ – g
3
,
℘

zz
= 6℘
2
–
1
2
g
2
.
18.14.2-4. Connection with Jacobi elliptic functions.
Direct and inverse representations of the Weierstrass elliptic function via Jacobi elliptic
functions:
℘(z)=e
1
+(e
1
– e
3
)

cn
2
w
sn
2
w
= e
2
+(e
1
– e
3
)
dn
2
w
sn
2
w
= e
3
+
e
1
– e
3
sn
2
w
;

sn w =

e
1
– e
3
℘(z)–e
3
,cnw =

℘(z)–e
1
℘(z)–e
3
,dnw =

℘(z)–e
2
℘(z)–e
3
;
w = z
√
e
1
– e
3
= K z/ω
1
.

The parameters are related by
k =

e
2
– e
3
e
1
– e
3
, k

=

e
1
– e
2
e
1
– e
3
, K = ω
1
√
e
1
– e
3

, i K

= ω
2
√
e
1
– e
3
.
978 SPECIAL FUNCTIONS AND THEIR PROPERTIES
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:
ϑ
1
(v)=ϑ
1
(v, q)=ϑ
1
(v|τ)=2
∞

n=0
(–1)
n
q
(n+1/2)

2
sin[(2n + 1)πv]=i
∞

n=–∞
(–1)
n
q
(n–1/2)
2
e
iπ(2n–1)v
,
ϑ
2
(v)=ϑ
2
(v, q)=ϑ
2
(v|τ)=2
∞

n=0
q
(n+1/2)
2
cos[(2n + 1)πv]=
∞

n=–∞

q
(n–1/2)
2
e
iπ(2n–1)v
,
ϑ
3
(v)=ϑ
3
(v, q)=ϑ
3
(v|τ)=1 + 2
∞

n=0
q
n
2
cos(2nπv)=
∞

n=–∞
q
n
2
e
2iπnv
,
ϑ

4
(v)=ϑ
4
(v, q)=ϑ
4
(v|τ)=1 + 2
∞

n=0
(–1)
n
q
n
2
cos(2nπv)=
∞

n=–∞
(–1)
n
q
n
2
e
2iπnv
,
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:
ϑ
1
(v) odd, has period 2, vanishes at v = m + nτ;
ϑ
2
(v) even, has period 2, vanishes at v = m + nτ +
1
2
;
ϑ
3
(v) even, has period 1, vanishes at v = m +(n +
1
2
)τ +
1
2
;
ϑ
4
(v) even, has period 1, vanishes at v = m +(n +
1
2
)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):
ϑ
1

v +
1
2

= ϑ
2
(v), ϑ
2

v +
1
2

=–ϑ
1
(v),
ϑ
3

v +
1
2


= ϑ
4
(v), ϑ
4

v +
1
2

= ϑ
3
(v),
ϑ
1

v +
τ
2

= ie
–iπ

v+
τ
4

ϑ
4
(v), ϑ

2

v +
τ
2

= e
–iπ

v+
τ
4

ϑ
3
(v),
ϑ
3

v +
τ
2

= e
–iπ

v+
τ
4


ϑ
2
(v), ϑ
4

v +
τ
2

= ie
–iπ

v+
τ
4

ϑ
1
(v).
18.15. JACOBI THETA FUNCTIONS 979
Linear relations (second set):
ϑ
1
(v|τ + 1)=e
iπ/4
ϑ
1
(v|τ), ϑ
2
(v|τ + 1)=e

iπ/4
ϑ
2
(v|τ),
ϑ
3
(v|τ + 1)=ϑ
4
(v|τ), ϑ
4
(v|τ + 1)=ϑ
3
(v|τ),
ϑ
1

v
τ



–
1
τ

=
1
i

τ

i
e
iπv
2
/τ
ϑ
1
(v|τ), ϑ
2

v
τ



–
1
τ

=

τ
i
e
iπv
2
/τ
ϑ
4
(v|τ),

ϑ
3

v
τ



–
1
τ

=

τ
i
e
iπv
2
/τ
ϑ
3
(v|τ), ϑ
4

v
τ




–
1
τ

=

τ
i
e
iπv
2
/τ
ϑ
2
(v|τ).
Quadratic relations:
ϑ
2
1
(v)ϑ
2
2
(0)=ϑ
2
4
(v)ϑ
2
3
(0)–ϑ
2

3
(v)ϑ
2
4
(0),
ϑ
2
1
(v)ϑ
2
3
(0)=ϑ
2
4
(v)ϑ
2
2
(0)–ϑ
2
2
(v)ϑ
2
4
(0),
ϑ
2
1
(v)ϑ
2
4

(0)=ϑ
2
3
(v)ϑ
2
2
(0)–ϑ
2
2
(v)ϑ
2
3
(0),
ϑ
2
4
(v)ϑ
2
4
(0)=ϑ
2
3
(v)ϑ
2
3
(0)–ϑ
2
2
(v)ϑ
2

2
(0).
18.15.2-2. Representation of the theta functions in the form of infinite products.
ϑ
1
(v)=2q
0
q
1/4
sin(πv)
∞

n=1

1 – 2q
2n
cos(2πv)+q
4n

,
ϑ
2
(v)=2q
0
q
1/4
cos(πv)
∞

n=1


1 + 2q
2n
cos(2πv)+q
4n

,
ϑ
3
(v)=q
0
∞

n=1

1 + 2q
2n–1
cos(2πv)+q
4n–2

,
ϑ
4
(v)=q
0
∞

n=1

1 – 2q

2n–1
cos(2πv)+q
4n–2

,
where q
0
=
∞

n=1
(1 – q
2n
).
18.15.2-3. Connection with Jacobi elliptic functions.
Representations of Jacobi elliptic functions in terms of the theta functions:
sn w =
ϑ
3
(0)
ϑ
2
(0)
ϑ
1
(v)
ϑ
4
(v)
,cnw =

ϑ
4
(0)
ϑ
2
(0)
ϑ
2
(v)
ϑ
4
(v)
,dnw =
ϑ
4
(0)
ϑ
3
(0)
ϑ
3
(v)
ϑ
4
(v)
, w = 2 K v.
The parameters are related by
k =
ϑ
2

2
(0)
ϑ
2
3
(0)
, k

=
ϑ
2
4
(0)
ϑ
2
3
(0)
, K =
π
2
ϑ
2
3
(0), K

=–iτ K.
980 SPECIAL FUNCTIONS AND THEIR PROPERTIES
TABLE 18.6
The Mathieu functions ce
n

=ce
n
(x, q)andse
n
=se
n
(x, q) (for odd n, functions
ce
n
and se
n
are 2π-periodic, and for even n,theyareπ-periodic); definite
eigenvalues a = a
n
(q)anda = b
n
(q) correspond to each value of parameter q
Mathieu functions
Recurrence relations
for coefficients
Normalization
conditions
ce
2n
=
∞

m=0
A
2n

2m
cos 2mx
qA
2n
2
= a
2n
A
2n
0
;
qA
2n
4
=(a
2n
–4)A
2n
2
–2qA
2n
0
;
qA
2n
2m+2
=(a
2n
–4m
2

)A
2n
2m
–qA
2n
2m–2
, m ≥ 2
(A
2n
0
)
2
+
∞

m=0
(A
2n
2m
)
2
=

2 if n = 0
1 if n ≥ 1
ce
2n+1
=
∞


m=0
A
2n+1
2m+1
cos(2m+1)x
qA
2n+1
3
=(a
2n+1
–1–q)A
2n+1
1
;
qA
2n+1
2m+3
=[a
2n+1
–(2m+1)
2
]A
2n+1
2m+1
–qA
2n+1
2m–1
, m ≥ 1
∞


m=0
(A
2n+1
2m+1
)
2
= 1
se
2n
=
∞

m=0
B
2n
2m
sin 2mx,
se
0
= 0
qB
2n
4
=(b
2n
–4)B
2n
2
;
qB

2n
2m+2
=(b
2n
–4m
2
)B
2n
2m
–qB
2n
2m–2
, m ≥ 2
∞

m=0
(B
2n
2m
)
2
= 1
se
2n+1
=
∞

m=0
B
2n+1

2m+1
sin(2m+1)x
qB
2n+1
3
=(b
2n+1
–1–q)B
2n+1
1
;
qB
2n+1
2m+3
=[b
2n+1
–(2m+1)
2
]B
2n+1
2m+1
–qB
2n+1
2m–1
, m ≥ 1
∞

m=0
(B
2n+1

2m+1
)
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 ce
n
(x, q)andse
n
(x, q) are periodical solutions of the Mathieu
equation
y

xx
+(a – 2q cos 2x)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:
ce
2n
(x,–q)=(–1)
n
ce
2n

π

2
–x, q

,ce
2n+1
(x,–q)=(–1)
n
se
2n+1

π
2
–x, q

,
se
2n
(x,–q)=(–1)
n–1
se
2n

π
2
–x, q

,se
2n+1
(x,–q)=(–1)
n

ce
2n+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
n
m
)to
zero, we obtain an algebraic equation for finding a
n
(q)(orb
n
(q)).
18.16. MAT H I E U FUNCTIONS AND MODIFIED MAT H I EU FUNCTIONS 981
For fixed real q ≠ 0, eigenvalues a
n

and b
n
are all real and different, while
if q > 0 then a
0
< b
1
< a
1
< b
2
< a
2
< ···;
if q < 0 then a
0
< a
1
< b
1
< b
2
< a
2
< a
3
< b
3
< b
4

< ··· .
The eigenvalues possess the properties
a
2n
(–q)=a
2n
(q), b
2n
(–q)=b
2n
(q), a
2n+1
(–q)=b
2n+1
(q).
Tables of the eigenvalues a
n
= a
n
(q)andb
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
)hasn zeros

on the interval 0 ≤ x < π (q is a real number).
18.16.1-3. Asymptotic expansions as q → 0 and q →∞.
Listed below are two leading terms of asymptotic expansions of the Mathieu functions
ce
n
(x, q)andse
n
(x, q), as well as of the corresponding eigenvalues a
n
(q)andb
n
(q), as
q → 0:
ce
0
(x, q)=
1
√
2

1 –
q
2
cos 2x

, a
0
(q)=–
q
2

2
+
7q
4
128
;
ce
1
(x, q)=cosx –
q
8
cos 3x, a
1
(q)=1 + q;
ce
2
(x, q)=cos2x +
q
4

1 –
cos 4x
3

, a
2
(q)=4 +
5q
2
12

;
ce
n
(x, q)=cosnx +
q
4

cos(n + 2)x
n + 1
–
cos(n – 2)x
n – 1

, a
n
(q)=n
2
+
q
2
2(n
2
– 1)
(n ≥ 3);
se
1
(x, q)=sinx –
q
8
sin 3x, b

1
(q)=1 – q;
se
2
(x, q)=sin2x – q
sin 4x
12
, b
2
(q)=4 –
q
2
12
;
se
n
(x, q)=sinnx –
q
4

sin(n + 2)x
n + 1
–
sin(n – 2)x
n – 1

, b
n
(q)=n
2

+
q
2
2(n
2
– 1)
(n ≥ 3).
Asymptotic results as q →∞(–π/2 < x < π/2):
a
n
(q) ≈ –2q + 2(2n + 1)
√
q +
1
4
(2n
2
+ 2n + 1),
b
n+1
(q) ≈ –2q + 2(2n + 1)
√
q +
1
4
(2n
2
+ 2n + 1),
ce
n

(x, q) ≈ λ
n
q
–1/4
cos
–n–1
x

cos
2n+1
ξ exp(2
√
q sin x)+sin
2n+1
ξ exp(–2
√
q sin x)

,
se
n+1
(x, q) ≈ μ
n+1
q
–1/4
cos
–n–1
x

cos

2n+1
ξ exp(2
√
q sin x)–sin
2n+1
ξ exp(–2
√
q sin x)

,
where λ
n
and μ
n
are some constants independent of the parameter q,and ξ =
1
2
x +
π
4
.
982 SPECIAL FUNCTIONS AND THEIR PROPERTIES
18.16.2. Modified Mathieu Functions
The modified Mathieu functions Ce
n
(x, q)andSe
n
(x, q) are solutions of the modified
Mathieu equation
y


xx
–(a – 2q cosh 2x)y = 0,
with a = a
n
(q)anda = b
n
(q) being the eigenvalues of the Mathieu equation (see Subsection
18.16.1).
The modified Mathieu functions are defined as
Ce
2n+p
(x, q)=ce
2n+p
(ix, q)=
∞

k=0
A
2n+p
2k+p
cosh[(2k + p)x],
Se
2n+p
(x, q)=–i se
2n+p
(ix, q)=
∞

k=0

B
2n+p
2k+p
sinh[(2k + p)x],
where p may be equal to 0 and 1, and coefficients A
2n+p
2k+p
and B
2n+p
2k+p
are indicated in
Subsection 18.16.1.
18.17. Orthogonal Polynomials
All zeros of each of the orthogonal polynomials P
n
(x) considered in this section are real
and simple. The zeros of the polynomials P
n
(x)andP
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
– n
2
x
n–1
+
n
2
(n – 1)
2
2!
x
n–2
+ ···

.
The first four polynomials have the form
L
0
(x)=1, L
1
(x)=–x + 1, L
2
(x)=
1
2
(x
2

– 4x + 2), L
3
(x)=
1
6
(–x
3
+ 9x
2
– 18x + 6).
To calculate L
n
(x)forn ≥ 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
weight e

–x
:

∞
0
e
–x
L
n
(x)L
m
(x) dx =

0 if n ≠ m,
1 if n = m.
The generating function is
1
1 – s
exp

–
sx
1 – s

=
∞

n=0
L
n

(x)s
n
, |s| < 1.