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28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.6 Expansions for Small q

Contents
  1. §28.6(i) Eigenvalues
  2. §28.6(ii) Functions cen and sen

§28.6(i) Eigenvalues

Leading terms of the power series for am(q) and bm(q) for m6 are:

28.6.1 a0(q) =12q2+7128q4292304q6+68687188 74368q8+,
28.6.2 a1(q) =1+q18q2164q311536q4+1136864q5+495 89824q6+5594 37184q783353 89440q8+,
28.6.3 b1(q) =1q18q2+164q311536q41136864q5+495 89824q65594 37184q783353 89440q8+,
28.6.4 a2(q) =4+512q276313824q4+10 02401796 26240q616690 6840145 86471 42400q8+,
28.6.5 b2(q) =4112q2+513824q4289796 26240q6+2139145 86471 42400q8+,
28.6.6 a3(q) =9+116q2+164q3+1320480q4516384q51961235 92960q66091048 57600q7+,
28.6.7 b3(q) =9+116q2164q3+1320480q4+516384q51961235 92960q6+6091048 57600q7+,
28.6.8 a4(q) =16+130q2+4338 64000q4570127216 00000q6+,
28.6.9 b4(q) =16+130q23178 64000q4+1004927216 00000q6+,
28.6.10 a5(q) =25+148q2+117 74144q4+11 47456q5+378918 13888q6+,
28.6.11 b5(q) =25+148q2+117 74144q411 47456q5+378918 13888q6+,
28.6.12 a6(q) =36+170q2+187439 04000q4+67 436179293 59872 00000q6+,
28.6.13 b6(q) =36+170q2+187439 04000q458 616339293 59872 00000q6+.

Leading terms of the of the power series for m=7,8,9, are:

28.6.14 am(q)bm(q)}=m2+12(m21)q2+5m2+732(m21)3(m24)q4+9m4+58m2+2964(m21)5(m24)(m29)q6+.

For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).

The coefficients of the power series of a2n(q), b2n(q) and also a2n+1(q), b2n+1(q) are the same until the terms in q2n2 and q2n, respectively. Then

28.6.15 am(q)bm(q)=2qm(2m1(m1)!)2(1+O(q2)).

Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations:

28.6.16 a(2n)2q2a(2n2)2q2a(2n4)2q2a222q2a=q2(2n+2)2aq2(2n+4)2a,
a=a2n(q),
28.6.17 a(2n+1)2q2a(2n1)2q2a32q2a12q=q2(2n+3)2aq2(2n+5)2a,
a=a2n+1(q),
28.6.18 a(2n+1)2q2a(2n1)2q2a32q2a12+q=q2(2n+3)2aq2(2n+5)2a,
a=b2n+1(q),
28.6.19 a(2n+2)2q2a(2n)2q2a(2n2)2q2a22=q2(2n+4)2aq2(2n+6)2a,
a=b2n+2(q).

Numerical values of the radii of convergence ρn(j) of the power series (28.6.1)–(28.6.14) for n=0,1,,9 are given in Table 28.6.1. Here j=1 for a2n(q), j=2 for b2n+2(q), and j=3 for a2n+1(q) and b2n+1(q). (Table 28.6.1 is reproduced from Meixner et al. (1980, §2.4).)

Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.
n ρn(1) ρn(2) ρn(3)
0 or 1 1.46876 86138 6.92895 47588 3.76995 74940
2 7.26814 68935 16.80308 98254 11.27098 52655
3 16.47116 58923 30.09677 28376 22.85524 71216
4 30.42738 20960 48.13638 18593 38.52292 50099
5 47.80596 57026 69.59879 32769 58.27413 84472
6 69.92930 51764 95.80595 67052 82.10894 36067
7 95.47527 27072 125.43541 1314 110.02736 9210
8 125.76627 89677 159.81025 4642 142.02943 1279
9 159.47921 26694 197.60667 8692 178.11513 940

It is conjectured that for large n, the radii increase in proportion to the square of the eigenvalue number n; see Meixner et al. (1980, §2.4). It is known that

28.6.20 lim infnρn(j)n2kk(K(k))2=2.04183 4,

where k is the unique root of the equation 2E(k)=K(k) in the interval (0,1), and k=1k2. For E(k) and K(k) see §19.2(ii).

§28.6(ii) Functions cen and sen

Leading terms of the power series for the normalized functions are:

28.6.21 21/2ce0(z,q) =112qcos2z+132q2(cos4z2)1128q3(19cos6z11cos2z)+,
28.6.22 ce1(z,q) =cosz18qcos3z+1128q2(23cos5z2cos3zcosz)11024q3(19cos7z89cos5z13cos3z+2cosz)+,
28.6.23 se1(z,q) =sinz18qsin3z+1128q2(23sin5z+2sin3zsinz)11024q3(19sin7z+89sin5z13sin3z2sinz)+,
28.6.24 ce2(z,q) =cos2z14q(13cos4z1)+1128q2(13cos6z769cos2z)+,
28.6.25 se2(z,q) =sin2z112qsin4z+1128q2(13sin6z49sin2z)+.

For m=3,4,5,,

28.6.26 cem(z,q)=cosmzq4(1m+1cos(m+2)z1m1cos(m2)z)+q232(1(m+1)(m+2)cos(m+4)z+1(m1)(m2)cos(m4)z2(m2+1)(m21)2cosmz)+.

For the corresponding expansions of sem(z,q) for m=3,4,5, change cos to sin everywhere in (28.6.26). For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).’

The radii of convergence of the series (28.6.21)–(28.6.26) are the same as the radii of the corresponding series for an(q) and bn(q); compare Table 28.6.1 and (28.6.20).