Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 113, 11-15 (2008)]
On the Asymptotic Behavior of the Fourier Coefficients of Mathieu Functions
Volume 113 Gerhard Wolf Universität Duisburg-Essen Campus Essen, Fachbereich Mathematik D 45110 Essen, Germany
Number 1 The asymptotic behavior of the Fourier n n coefficients Am (q) and Bm (q) of the periodic Mathieu functions cen(z, q) and sen(z, q) is derived for fixed n and q (≠0), as m → ∞. Error bounds can be constructed for all approximations.
January-February 2008 Key words: asymptotic behavior; Fourier coefficients; Mathieu functions. Accepted: January 2, 2008 Available online: http://www.nist.gov/jres
[email protected]
The DLMF is modeled after the enormously successful but increasingly out-of-date NBS Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 55, M. Abramowitz and I. A. Stegun (editors), 1964. The NBS handbook has sold more than 700,000 copies and is frequently cited in scientific journal articles. The need for a modern reference is being filled by NIST editors and staff, aided by the scientific content provided by approximately 50 external authors and validators In addition to the main purpose of the DLMF, which is to provide a comprehensive and authoritative research tool, the project also seeks to guide further research in special functions. The paper that follows is an example. It provides the proofs of results that Wolf presents for the first time in §28.4(vii) of the DLMF.
Foreword Mathieu functions have many applications, especially in mathematics and physics: • separation of variables in elliptical coordinates • separation of variables in parabolic coordinates • vibrations in a stretched elliptical ring membrane • stability of a pendulum with periodically varying length • amplitude distortion in moving-coil loudspeakers • relativistic oscillators These applications are mentioned in the chapter on Mathieu functions, written by Gerhard Wolf for the NIST Digital Library of Mathematical Functions. Professor Wolf is coauthor (with J. Meixner and F. W. Schäfke) of Mathieu Functions and Spheroidal Functions and their Mathematical Foundations, Lecture Notes in Mathematics 837, Springer-Verlag, 1980. The DLMF is scheduled to begin service in 2009 from a NIST Web site. A hardcover book will be published also. These resources will provide a comprehensive guide to the higher mathematical functions for use by experienced scientific professionals.
Daniel W. Lozier DLMF General Editor NIST Mathematical and Computational Sciences Division
11
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology 1.
Introduction and Definitions
(a − 4 m2 ) B2 m − q( B2 m − 2 + B2 m + 2 ) = 0,
In this paper we aim to give asymptotic formulae for Fourier coefficients of the periodic solutions of Mathieu’s equation w′′ + ( a − 2 q cos 2 z) w = 0.
In §2 we examine the asymptotic behavior of the coef2n 2n+1 2n+1 2n+2 ficients A2m (q), A2m+1 (q), B2m+1 (q), and B2m+2 (q) for fixed n, q (≠ 0), as m → ∞.
(1)
wII (0; a, q) ⎤ ⎡1 0 ⎤ . = wII′ (0; a, q) ⎥⎦ ⎢⎣0 1 ⎥⎦
2.
A22mn ( q) (−1) m = A02 n ( q) ( m!) 2
a = a2 n ( q), A2 m = A22mn ( q),
(a −1 − q) A1 − qA3 = 0,
m = 2,3, 4,..., (3) 2 n+1 2 m+1
a = a2 n+1 ( q), A2 m+1 = A
( q),
a = b2 n+1 ( q), B2 m+1 = B
⎛q⎞ ⎜ ⎟ ⎝4⎠
m +1
2(1 + O(1/ m)) , (8) wII′ ( 12 π ; a2 n +1 ( q), q)
B22mn ++11 ( q) (−1) m = 1 2 n +1 B1 ( q) (( 2 ) m +1 ) 2
⎛q⎞ ⎜ ⎟ ⎝4⎠
m +1
2(1 + O(1/ m)) , wI ( 12 π ; b2 n +1 ( q), q)
⎛ a zm +1 = ⎜1 − 2 ⎝ (2 m)
(a − (2 m +1) ) A2 m+1 − q( A2 m−1 + A2 m+3 ) = 0, m =1,2,3,..., (4) (a −1 + q) B1 − qB3 = 0,
(7)
(9)
m
⎛ q ⎞ qπ (1 + O(1/ m)) . ⎜ ⎟ ′ 1 ⎝ 4 ⎠ wI ( 2 π ; b2 n + 2 ( q), q)
(10)
Proof of (7). We consider for q ≠ 0 the three termrecurrence relation
2
2 n+1 2 m+1
m
⎛ q ⎞ π (1 + O(1/ m)) , ⎜ ⎟ ⎝ 4 ⎠ wII ( 12 π ; a2 n ( q), q)
A22mn ++11 ( q) (−1) m +1 = 1 2 n +1 A1 ( q) (( 2 ) m +1 ) 2
B22mn + 2 ( q) (−1) m = B22 n + 2 ( q) ( m!) 2
(a − 4) A2 − q(2 A0 + A4 ) = 0, (a − 4 m2 ) A2 m − q( A2 m − 2 + A2 m + 2 ) = 0,
Asymptotic Forms
The following result will be proved: Proposition 1. For fixed n and q ≠ 0, as m → ∞
(2)
Furthermore, we obtain eigenvalues and eigenfunctions of (1) for n = 0, 1, 2, ... . Table 1 gives their notations and properties. “Period π” means that the eigenfunction has the property w(z + π) = w(z), whereas “Antiperiod π” means that w(z + π) = –w(z). “Even parity” means w(–z) = w(z) and “Odd parity” means w(–z) = –w(z). The Fourier coefficients satisfy the recurrence relations aA0 − qA2 = 0,
m = 2,3, 4,... .
(6)
Equation (1) possesses the fundamental pair of solutions wI(z; a, q), wII(z; a, q) called basic solutions (see Ref. [1]) with ⎡ wI (0; a, q) ⎢ w′ (0; a, q) ⎣ I
a = b2 n + 2 ( q), B2 m + 2 = B22mn ++22 ( q)
(a − 4) B2 − qB4 = 0,
( q),
⎞ q2 − z zm −1 , ⎟ m 16 m2 ( m − 1) 2 ⎠
m = 2,3,... .
(11)
For the two independent solutions um, vm of (11) with u1 = a, u2 = a (1 – 14 a) + 12 q2 and v1 = 1, v2 = (1 – 14 a), it
2
(a − (2 m +1) ) B2 m+1 − q( B2 m−1 + B2 m+3 ) = 0, m =1,2,3,..., (5)
Table 1. Eigenvalues and eigenfunctions Eigenvalues
Eigenfunctions
Periodicity
Parity
a = a2n(q)
ce2n(z, q)
Period π
Even
∑
∞
a = a2n+1(q)
ce2n+1(z, q)
Antiperiod π
Even
∑
∞
a = b2n+1(q)
se2n+1(z, q)
Antiperiod π
Odd
∑
∞
a = b2n+2(q)
se2n+2(z, q)
Period π
Odd
∑
∞
12
Fourier series m=0
A22mn (q ) cos(2 m) z
m =0
m=0
m=0
A22mn ++11 (q ) cos(2 m +1) z B22mn ++11 (q ) sin(2 m +1) z B22mn ++22 (q ) sin(2 m + 2) z
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology ⎛ a2 n (q) ⎞ q2 1 1 ρ ρ ρm +1 , − − = ⎜ m −1 2 ⎟ m 2 16 (2 ) ( m m m + 1) 2 ⎝ ⎠
follows from Sätze (Theorems) 1 and 3 of F. W. Schäfke [2] that lim um = −(2 / π ) wI′((π / 2); a, q),
m →∞
lim vm = (2 / π ) wII ((π / 2); a, q).
m →∞
ρ m − ρm −1 = O (1/ m 2 ), and ρm = k + O (1/ m).
(12)
(21)
(22)
The constant k is determined with the aid of (17): Transformation of (11) with C2 m = (−1)
m −1
4 m −1 (( m −1)!) 2 zm , m = 1, 2,3,..., m −1 q
U 2( m +1)V2 m − V2( m +1) U2 m
(13)
⎛ q⎞ = ⎜− ⎟ ⎝ 4⎠
m +1
1 4 m −1 ρ m +1 ( −1) m −1 m −1 (( m −1)!) 2 vm 2 ((m + 1)!) q m
4m 1 ⎛ q⎞ ρm − ( −1) m ( m!) 2 vm +1 ⎜ − ⎟ 2 4 q ⎝ ⎠ ( m!)
yields
m
q (C2( m +1) + C2( m −1) ) = ( a − 4 m2 ) C2 m , m = 2,3,... .
(14)
2
1 ⎛q⎞ =⎜ ⎟ ρ m +1vm − vm +1 ρm = −2 q, 4 ( ( − 1)) 2 m m ⎝ ⎠
(23)
We note the following special solutions of (14): U 2 m with zm = um , V2 m with zm = vm.
and for m → ∞
(15)
(2 k / π ) wII ( 12 π ; a2 n ( q), q) = 2 q.
Then
Together with (19) we obtain (7). Proof of (10). In the same way it follows, if a = b2n+2(q), then wII ( 12 π; a, q) = 0, vm → 0, and
U 2 = a, U 4 = −(4/ q)( a(1 − 14 a) + 12 q2 ), V2 = 1,
V4 = −(4/ q)(1 − 14 a).
(16)
(a − 4)V2 − qV4 = 0,
Furthermore, we have
2n+2 Comparison with B2m (q) of (6) shows that
B22mn + 2 ( q) = V2 m . B22 n + 2 ( q)
aU 0 − qU 2 = 0, ( a − 4)U 2 − q(2U0 + U4 ) = 0, 2
q(U 2( m +1) + U2( m −1) ) = ( a − 4 m ) U2 m .
(18)
m
1 ⎛ q⎞ ρm V2 m = ⎜ − ⎟ 2 ⎝ 4 ⎠ (m !)
(27)
we find again ρm – ρm–1 = O(1/m2) and (19)
ρ m = k + O(1/ m).
Thus U2m is the minimal solution of (14), and by the substitution
(28)
The constant k can be computed via (17): U 2( m +1)V2 m − V2( m +1) U2 m
m
1 ⎛ q⎞ ρm = ⎜− ⎟ 2 ⎝ 4 ⎠ (m !)
(26)
Thus V2m is the minimal solution of (14), and for
2n Comparison with A2m (q) of (3) shows that
A2 n ( q) = U 2m . q 22mn A0 ( q)
(25)
(17)
If, now, a = a2n(q), then w′I(π/2; a2n(q), q) = 0 and um → 0 as m → ∞. Set U0 = q. Then for m = 2, 3, 4, ...,
U 2m
q(V2( m +1) + V2( m −1) ) = ( a − 4 m2 ) V2 m ,
m = 2,3, 4,... .
U 2( m +1)V2 m − V2( m +1) U2 m = U4 V2 − V4 U2 = −2 q.
(24)
2
(20)
1 ⎛q⎞ = um +1 ρ m − ⎜ ⎟ ρm +1 um 2 ⎝ 4 ⎠ (m( m + 1)) = −2q.
we find that
Letting m → ∞, we obtain 13
(29)
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology
k=
qπ . 1 wI′ ( 2 π ; b2 n + 2 ( q), q)
(30)
q
⎛ q⎞ U 2 m +1 = ⎜ − ⎟ ⎝ 4⎠
⎛ ⎞ a = ⎜1 − z 2 ⎟ m ⎝ (2m + 1) ⎠ q2 − zm −1 , m = 1, 2,3,... . 16( m + 12 ) 2 ( m − 12 ) 2 (31)
m +1
1
((
)
2 1 2 m +1
)
ρm
(39)
we find that ⎛ a2 n +1 ( q) ⎞ q2 ρ ρ ρm +1 , − = ⎜1 − ⎟ m m −1 2 (2 m +1) 2 (2 m + 3) 2 ⎝ (2 m + 1) ⎠
The two independent solutions um and vm of (31) with u0 = 1, u1 = q – a + 1 and v0 = 1, v1 = –q – a + 1, respectively, satisfy
(40)
ρ m − ρm −1 = O (1/ m2 ),
lim um = wI (π / 2; a, q),
ρm = k + O(1/ m).
(41)
The constant k is determined with the aid of (37):
m →∞
lim vm = wII′ (π / 2; a, q);
(32)
m →∞
U 2 m +1V2 m −1 − V2 m +1U2 m −1 = O(1/ m2 ) − qvm ρm −1 = −2 q2 . (42)
see Sätze (Theorems) 1 and 3 of F. W. Schäfke [2]). Then transformation of (31) with1 C2 m +1 = ( −4 / q) m q(( 12 ) m ) 2 zm ,
(38)
Thus U2m+1 is the minimal solution of (34), and with
Together with (26) we obtain the formula (10). Proof of (8) and (9). We start with the recurrence relations zm +1
A22mn ++11 ( q) = U 2 m +1 . A12 n +1 ( q)
m = 0,1, 2,3,...,
Letting m → ∞, we obtain k=
(33)
2q . wII′ (π / 2; a2 n +1 ( q), q)
(43)
yields q (C2 m + 3 + C2 m −1 ) = ( a − (1 + 2 m) 2 ) C2 m +1 ,
Together with (38) we find (8). The formula (9) is obtained by the transformation q → –q, a2n+1(–q) = b2n+1(q), and A2n+1 2m+1(–q) = 2n+1 (–1)n–mB2m+1 (q).
m = 1, 2,3,... . (34)
We note the special solutions U 2 m +1 with zm = um , V2 m +1 with zm = vm ,
(35)
3.
Improvement of the Rate of Convergence
(36)
If a2n ≠ (2m)2, then for m = 0, 1, 2, . . . we transform (21) via
and obtain U1 = q, U3 = a − 1 − q, V1 = q, V3 = a −1 + q.
Furthermore, we observe that
ρm = ζ m 2
U 2 m +1V2 m −1 − V2 m +1U2 m −1 = U3 V1 − V3 U1 = −2 q .
m
⎛
a2 n ( q ) ⎞ , 2 ⎟ ⎠
∏ ⎜⎝1 − (2κ ) κ =1
(37)
(44)
and obtain (as in Ref. [2]) ζm – ζm–1 = O(1/m4), ζm – ζ = O(1/m3), where ζ = 12 π sin 12 π a2 n ( q) k. Finally we have
(
If, now, a = a2n+1(q), then wI(π/2; a2n+1(q), q) = 0 and um → 0 for m → ∞. 2n+1 Comparison with A2m+1 (q) of (4) shows that
A22mn ( q) (−1) m = A02 n ( q) ( m!) 2
m
⎛q⎞ ⎜ ⎟ × ⎝4⎠ 2sin
1
( 12 )m means the Pochhammer symbol: (a)n = a(a + 1)(a + 2) . . . (a + n – 1).
)
(π 1 2
)
a2 n ( q) (1 + O(1/ m3 ))
. a2 n ( q) ∏ (1 − (2κ ) −2 a2 n ( q)) wII ( 12 π; a2 n ( q), q) κ =1 (45) m
14
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology In the other cases, for example (40), similar results can be obtained. Error bounds are possible with the aid of Refs. [2] and [3]. Remark: Further improvements of the rate of convergence are possible by application of the results of Ref. [3].
4.
References
[1] J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme, Springer (1954). [2] F. W. Schäfke, Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I, Numer. Math. 3, 30-38 (1961). [3] F. W. Schäfke, R. Ebert, and H. Groh, Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung II, Numer. Math. 4, 1-7 (1962).
About the author: Gerhard Wolf is a mathematician at the Universität Duisburg-Essen, Germany. He is the author of the chapter on Mathieu Functions and Hill’s Equation in the NIST Digital Library of Mathematical Functions. The National Institute of Standards and Technology is an agency of the U.S. Department of Commerce.
15
Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 113, 11-15 (2008)]
On the Asymptotic Behavior of the Fourier Coefficients of Mathieu Functions
Volume 113 Gerhard Wolf Universität Duisburg-Essen Campus Essen, Fachbereich Mathematik D 45110 Essen, Germany
Number 1 The asymptotic behavior of the Fourier n n coefficients Am (q) and Bm (q) of the periodic Mathieu functions cen(z, q) and sen(z, q) is derived for fixed n and q (≠0), as m → ∞. Error bounds can be constructed for all approximations.
January-February 2008 Key words: asymptotic behavior; Fourier coefficients; Mathieu functions. Accepted: January 2, 2008 Available online: http://www.nist.gov/jres
[email protected]
The DLMF is modeled after the enormously successful but increasingly out-of-date NBS Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 55, M. Abramowitz and I. A. Stegun (editors), 1964. The NBS handbook has sold more than 700,000 copies and is frequently cited in scientific journal articles. The need for a modern reference is being filled by NIST editors and staff, aided by the scientific content provided by approximately 50 external authors and validators In addition to the main purpose of the DLMF, which is to provide a comprehensive and authoritative research tool, the project also seeks to guide further research in special functions. The paper that follows is an example. It provides the proofs of results that Wolf presents for the first time in §28.4(vii) of the DLMF.
Foreword Mathieu functions have many applications, especially in mathematics and physics: • separation of variables in elliptical coordinates • separation of variables in parabolic coordinates • vibrations in a stretched elliptical ring membrane • stability of a pendulum with periodically varying length • amplitude distortion in moving-coil loudspeakers • relativistic oscillators These applications are mentioned in the chapter on Mathieu functions, written by Gerhard Wolf for the NIST Digital Library of Mathematical Functions. Professor Wolf is coauthor (with J. Meixner and F. W. Schäfke) of Mathieu Functions and Spheroidal Functions and their Mathematical Foundations, Lecture Notes in Mathematics 837, Springer-Verlag, 1980. The DLMF is scheduled to begin service in 2009 from a NIST Web site. A hardcover book will be published also. These resources will provide a comprehensive guide to the higher mathematical functions for use by experienced scientific professionals.
Daniel W. Lozier DLMF General Editor NIST Mathematical and Computational Sciences Division
11
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology 1.
Introduction and Definitions
(a − 4 m2 ) B2 m − q( B2 m − 2 + B2 m + 2 ) = 0,
In this paper we aim to give asymptotic formulae for Fourier coefficients of the periodic solutions of Mathieu’s equation w′′ + ( a − 2 q cos 2 z) w = 0.
In §2 we examine the asymptotic behavior of the coef2n 2n+1 2n+1 2n+2 ficients A2m (q), A2m+1 (q), B2m+1 (q), and B2m+2 (q) for fixed n, q (≠ 0), as m → ∞.
(1)
wII (0; a, q) ⎤ ⎡1 0 ⎤ . = wII′ (0; a, q) ⎥⎦ ⎢⎣0 1 ⎥⎦
2.
A22mn ( q) (−1) m = A02 n ( q) ( m!) 2
a = a2 n ( q), A2 m = A22mn ( q),
(a −1 − q) A1 − qA3 = 0,
m = 2,3, 4,..., (3) 2 n+1 2 m+1
a = a2 n+1 ( q), A2 m+1 = A
( q),
a = b2 n+1 ( q), B2 m+1 = B
⎛q⎞ ⎜ ⎟ ⎝4⎠
m +1
2(1 + O(1/ m)) , (8) wII′ ( 12 π ; a2 n +1 ( q), q)
B22mn ++11 ( q) (−1) m = 1 2 n +1 B1 ( q) (( 2 ) m +1 ) 2
⎛q⎞ ⎜ ⎟ ⎝4⎠
m +1
2(1 + O(1/ m)) , wI ( 12 π ; b2 n +1 ( q), q)
⎛ a zm +1 = ⎜1 − 2 ⎝ (2 m)
(a − (2 m +1) ) A2 m+1 − q( A2 m−1 + A2 m+3 ) = 0, m =1,2,3,..., (4) (a −1 + q) B1 − qB3 = 0,
(7)
(9)
m
⎛ q ⎞ qπ (1 + O(1/ m)) . ⎜ ⎟ ′ 1 ⎝ 4 ⎠ wI ( 2 π ; b2 n + 2 ( q), q)
(10)
Proof of (7). We consider for q ≠ 0 the three termrecurrence relation
2
2 n+1 2 m+1
m
⎛ q ⎞ π (1 + O(1/ m)) , ⎜ ⎟ ⎝ 4 ⎠ wII ( 12 π ; a2 n ( q), q)
A22mn ++11 ( q) (−1) m +1 = 1 2 n +1 A1 ( q) (( 2 ) m +1 ) 2
B22mn + 2 ( q) (−1) m = B22 n + 2 ( q) ( m!) 2
(a − 4) A2 − q(2 A0 + A4 ) = 0, (a − 4 m2 ) A2 m − q( A2 m − 2 + A2 m + 2 ) = 0,
Asymptotic Forms
The following result will be proved: Proposition 1. For fixed n and q ≠ 0, as m → ∞
(2)
Furthermore, we obtain eigenvalues and eigenfunctions of (1) for n = 0, 1, 2, ... . Table 1 gives their notations and properties. “Period π” means that the eigenfunction has the property w(z + π) = w(z), whereas “Antiperiod π” means that w(z + π) = –w(z). “Even parity” means w(–z) = w(z) and “Odd parity” means w(–z) = –w(z). The Fourier coefficients satisfy the recurrence relations aA0 − qA2 = 0,
m = 2,3, 4,... .
(6)
Equation (1) possesses the fundamental pair of solutions wI(z; a, q), wII(z; a, q) called basic solutions (see Ref. [1]) with ⎡ wI (0; a, q) ⎢ w′ (0; a, q) ⎣ I
a = b2 n + 2 ( q), B2 m + 2 = B22mn ++22 ( q)
(a − 4) B2 − qB4 = 0,
( q),
⎞ q2 − z zm −1 , ⎟ m 16 m2 ( m − 1) 2 ⎠
m = 2,3,... .
(11)
For the two independent solutions um, vm of (11) with u1 = a, u2 = a (1 – 14 a) + 12 q2 and v1 = 1, v2 = (1 – 14 a), it
2
(a − (2 m +1) ) B2 m+1 − q( B2 m−1 + B2 m+3 ) = 0, m =1,2,3,..., (5)
Table 1. Eigenvalues and eigenfunctions Eigenvalues
Eigenfunctions
Periodicity
Parity
a = a2n(q)
ce2n(z, q)
Period π
Even
∑
∞
a = a2n+1(q)
ce2n+1(z, q)
Antiperiod π
Even
∑
∞
a = b2n+1(q)
se2n+1(z, q)
Antiperiod π
Odd
∑
∞
a = b2n+2(q)
se2n+2(z, q)
Period π
Odd
∑
∞
12
Fourier series m=0
A22mn (q ) cos(2 m) z
m =0
m=0
m=0
A22mn ++11 (q ) cos(2 m +1) z B22mn ++11 (q ) sin(2 m +1) z B22mn ++22 (q ) sin(2 m + 2) z
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology ⎛ a2 n (q) ⎞ q2 1 1 ρ ρ ρm +1 , − − = ⎜ m −1 2 ⎟ m 2 16 (2 ) ( m m m + 1) 2 ⎝ ⎠
follows from Sätze (Theorems) 1 and 3 of F. W. Schäfke [2] that lim um = −(2 / π ) wI′((π / 2); a, q),
m →∞
lim vm = (2 / π ) wII ((π / 2); a, q).
m →∞
ρ m − ρm −1 = O (1/ m 2 ), and ρm = k + O (1/ m).
(12)
(21)
(22)
The constant k is determined with the aid of (17): Transformation of (11) with C2 m = (−1)
m −1
4 m −1 (( m −1)!) 2 zm , m = 1, 2,3,..., m −1 q
U 2( m +1)V2 m − V2( m +1) U2 m
(13)
⎛ q⎞ = ⎜− ⎟ ⎝ 4⎠
m +1
1 4 m −1 ρ m +1 ( −1) m −1 m −1 (( m −1)!) 2 vm 2 ((m + 1)!) q m
4m 1 ⎛ q⎞ ρm − ( −1) m ( m!) 2 vm +1 ⎜ − ⎟ 2 4 q ⎝ ⎠ ( m!)
yields
m
q (C2( m +1) + C2( m −1) ) = ( a − 4 m2 ) C2 m , m = 2,3,... .
(14)
2
1 ⎛q⎞ =⎜ ⎟ ρ m +1vm − vm +1 ρm = −2 q, 4 ( ( − 1)) 2 m m ⎝ ⎠
(23)
We note the following special solutions of (14): U 2 m with zm = um , V2 m with zm = vm.
and for m → ∞
(15)
(2 k / π ) wII ( 12 π ; a2 n ( q), q) = 2 q.
Then
Together with (19) we obtain (7). Proof of (10). In the same way it follows, if a = b2n+2(q), then wII ( 12 π; a, q) = 0, vm → 0, and
U 2 = a, U 4 = −(4/ q)( a(1 − 14 a) + 12 q2 ), V2 = 1,
V4 = −(4/ q)(1 − 14 a).
(16)
(a − 4)V2 − qV4 = 0,
Furthermore, we have
2n+2 Comparison with B2m (q) of (6) shows that
B22mn + 2 ( q) = V2 m . B22 n + 2 ( q)
aU 0 − qU 2 = 0, ( a − 4)U 2 − q(2U0 + U4 ) = 0, 2
q(U 2( m +1) + U2( m −1) ) = ( a − 4 m ) U2 m .
(18)
m
1 ⎛ q⎞ ρm V2 m = ⎜ − ⎟ 2 ⎝ 4 ⎠ (m !)
(27)
we find again ρm – ρm–1 = O(1/m2) and (19)
ρ m = k + O(1/ m).
Thus U2m is the minimal solution of (14), and by the substitution
(28)
The constant k can be computed via (17): U 2( m +1)V2 m − V2( m +1) U2 m
m
1 ⎛ q⎞ ρm = ⎜− ⎟ 2 ⎝ 4 ⎠ (m !)
(26)
Thus V2m is the minimal solution of (14), and for
2n Comparison with A2m (q) of (3) shows that
A2 n ( q) = U 2m . q 22mn A0 ( q)
(25)
(17)
If, now, a = a2n(q), then w′I(π/2; a2n(q), q) = 0 and um → 0 as m → ∞. Set U0 = q. Then for m = 2, 3, 4, ...,
U 2m
q(V2( m +1) + V2( m −1) ) = ( a − 4 m2 ) V2 m ,
m = 2,3, 4,... .
U 2( m +1)V2 m − V2( m +1) U2 m = U4 V2 − V4 U2 = −2 q.
(24)
2
(20)
1 ⎛q⎞ = um +1 ρ m − ⎜ ⎟ ρm +1 um 2 ⎝ 4 ⎠ (m( m + 1)) = −2q.
we find that
Letting m → ∞, we obtain 13
(29)
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology
k=
qπ . 1 wI′ ( 2 π ; b2 n + 2 ( q), q)
(30)
q
⎛ q⎞ U 2 m +1 = ⎜ − ⎟ ⎝ 4⎠
⎛ ⎞ a = ⎜1 − z 2 ⎟ m ⎝ (2m + 1) ⎠ q2 − zm −1 , m = 1, 2,3,... . 16( m + 12 ) 2 ( m − 12 ) 2 (31)
m +1
1
((
)
2 1 2 m +1
)
ρm
(39)
we find that ⎛ a2 n +1 ( q) ⎞ q2 ρ ρ ρm +1 , − = ⎜1 − ⎟ m m −1 2 (2 m +1) 2 (2 m + 3) 2 ⎝ (2 m + 1) ⎠
The two independent solutions um and vm of (31) with u0 = 1, u1 = q – a + 1 and v0 = 1, v1 = –q – a + 1, respectively, satisfy
(40)
ρ m − ρm −1 = O (1/ m2 ),
lim um = wI (π / 2; a, q),
ρm = k + O(1/ m).
(41)
The constant k is determined with the aid of (37):
m →∞
lim vm = wII′ (π / 2; a, q);
(32)
m →∞
U 2 m +1V2 m −1 − V2 m +1U2 m −1 = O(1/ m2 ) − qvm ρm −1 = −2 q2 . (42)
see Sätze (Theorems) 1 and 3 of F. W. Schäfke [2]). Then transformation of (31) with1 C2 m +1 = ( −4 / q) m q(( 12 ) m ) 2 zm ,
(38)
Thus U2m+1 is the minimal solution of (34), and with
Together with (26) we obtain the formula (10). Proof of (8) and (9). We start with the recurrence relations zm +1
A22mn ++11 ( q) = U 2 m +1 . A12 n +1 ( q)
m = 0,1, 2,3,...,
Letting m → ∞, we obtain k=
(33)
2q . wII′ (π / 2; a2 n +1 ( q), q)
(43)
yields q (C2 m + 3 + C2 m −1 ) = ( a − (1 + 2 m) 2 ) C2 m +1 ,
Together with (38) we find (8). The formula (9) is obtained by the transformation q → –q, a2n+1(–q) = b2n+1(q), and A2n+1 2m+1(–q) = 2n+1 (–1)n–mB2m+1 (q).
m = 1, 2,3,... . (34)
We note the special solutions U 2 m +1 with zm = um , V2 m +1 with zm = vm ,
(35)
3.
Improvement of the Rate of Convergence
(36)
If a2n ≠ (2m)2, then for m = 0, 1, 2, . . . we transform (21) via
and obtain U1 = q, U3 = a − 1 − q, V1 = q, V3 = a −1 + q.
Furthermore, we observe that
ρm = ζ m 2
U 2 m +1V2 m −1 − V2 m +1U2 m −1 = U3 V1 − V3 U1 = −2 q .
m
⎛
a2 n ( q ) ⎞ , 2 ⎟ ⎠
∏ ⎜⎝1 − (2κ ) κ =1
(37)
(44)
and obtain (as in Ref. [2]) ζm – ζm–1 = O(1/m4), ζm – ζ = O(1/m3), where ζ = 12 π sin 12 π a2 n ( q) k. Finally we have
(
If, now, a = a2n+1(q), then wI(π/2; a2n+1(q), q) = 0 and um → 0 for m → ∞. 2n+1 Comparison with A2m+1 (q) of (4) shows that
A22mn ( q) (−1) m = A02 n ( q) ( m!) 2
m
⎛q⎞ ⎜ ⎟ × ⎝4⎠ 2sin
1
( 12 )m means the Pochhammer symbol: (a)n = a(a + 1)(a + 2) . . . (a + n – 1).
)
(π 1 2
)
a2 n ( q) (1 + O(1/ m3 ))
. a2 n ( q) ∏ (1 − (2κ ) −2 a2 n ( q)) wII ( 12 π; a2 n ( q), q) κ =1 (45) m
14
Volume 113, Number 1, January-February 2008
Journal of Research of the National Institute of Standards and Technology In the other cases, for example (40), similar results can be obtained. Error bounds are possible with the aid of Refs. [2] and [3]. Remark: Further improvements of the rate of convergence are possible by application of the results of Ref. [3].
4.
References
[1] J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme, Springer (1954). [2] F. W. Schäfke, Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I, Numer. Math. 3, 30-38 (1961). [3] F. W. Schäfke, R. Ebert, and H. Groh, Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung II, Numer. Math. 4, 1-7 (1962).
About the author: Gerhard Wolf is a mathematician at the Universität Duisburg-Essen, Germany. He is the author of the chapter on Mathieu Functions and Hill’s Equation in the NIST Digital Library of Mathematical Functions. The National Institute of Standards and Technology is an agency of the U.S. Department of Commerce.
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