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

§28.6 Expansions for Small q

Contents

§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+7128q4-292304q6+68687188 74368q8+,
28.6.2 a1(q) =1+q-18q2-164q3-11536q4+1136864q5+495 89824q6+5594 37184q7-83353 89440q8+,
28.6.3 b1(q) =1-q-18q2+164q3-11536q4-1136864q5+495 89824q6-5594 37184q7-83353 89440q8+,
28.6.4 a2(q) =4+512q2-76313824q4+10 02401796 26240q6-16690 6840145 86471 42400q8+,
28.6.5 b2(q) =4-112q2+513824q4-289796 26240q6+2139145 86471 42400q8+,
28.6.6 a3(q) =9+116q2+164q3+1320480q4-516384q5-1961235 92960q6-6091048 57600q7+,
28.6.7 b3(q) =9+116q2-164q3+1320480q4+516384q5-1961235 92960q6+6091048 57600q7+,
28.6.8 a4(q) =16+130q2+4338 64000q4-570127216 00000q6+,
28.6.9 b4(q) =16+130q2-3178 64000q4+1004927216 00000q6+,
28.6.10 a5(q) =25+148q2+117 74144q4+11 47456q5+378918 13888q6+,
28.6.11 b5(q) =25+148q2+117 74144q4-11 47456q5+378918 13888q6+,
28.6.12 a6(q) =36+170q2+187439 04000q4+67 436179293 59872 00000q6+,
28.6.13 b6(q) =36+170q2+187439 04000q4-58 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(m2-1)q2+5m2+732(m2-1)3(m2-4)q4+9m4+58m2+2964(m2-1)5(m2-4)(m2-9)q6+.

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 q2n-2 and q2n, respectively. Then

28.6.15 am(q)-bm(q)=2qm(2m-1(m-1)!)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)2-q2a-(2n-2)2-q2a-(2n-4)2-q2a-22-2q2a=-q2(2n+2)2-a-q2(2n+4)2-a-,
a=a2n(q),
28.6.17 a-(2n+1)2-q2a-(2n-1)2-q2a-32-q2a-12-q=-q2(2n+3)2-a-q2(2n+5)2-a-,
a=a2n+1(q),
28.6.18 a-(2n+1)2-q2a-(2n-1)2-q2a-32-q2a-12+q=-q2(2n+3)2-a-q2(2n+5)2-a-,
a=b2n+1(q),
28.6.19 a-(2n+2)2-q2a-(2n)2-q2a-(2n-2)2-q2a-22=-q2(2n+4)2-a-q2(2n+6)2-a-,
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=1-k2. 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) =1-12qcos2z+132q2(cos4z-2)-1128q3(19cos6z-11cos2z)+,
28.6.22 ce1(z,q) =cosz-18qcos3z+1128q2(23cos5z-2cos3z-cosz)-11024q3(19cos7z-89cos5z-13cos3z+2cosz)+,
28.6.23 se1(z,q) =sinz-18qsin3z+1128q2(23sin5z+2sin3z-sinz)-11024q3(19sin7z+89sin5z-13sin3z-2sinz)+,
28.6.24 ce2(z,q) =cos2z-14q(13cos4z-1)+1128q2(13cos6z-769cos2z)+,
28.6.25 se2(z,q) =sin2z-112qsin4z+1128q2(13sin6z-49sin2z)+.

For m=3,4,5,,

28.6.26 cem(z,q)=cosmz-q4(1m+1cos(m+2)z-1m-1cos(m-2)z)+q232(1(m+1)(m+2)cos(m+4)z+1(m-1)(m-2)cos(m-4)z-2(m2+1)(m2-1)2cosmz)+.

For the corresponding expansions of sem(z,q) for m=3,4,5, change cos to sin everywhere in (28.6.26).

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).