# §28.6 Expansions for Small

## §28.6(i) Eigenvalues

Leading terms of the power series for and for are:

28.6.1

Leading terms of the of the power series for are:

The coefficients of the power series of , and also , are the same until the terms in and , respectively. Then

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

Numerical values of the radii of convergence of the power series (28.6.1)–(28.6.14) for are given in Table 28.6.1. Here for , for , and for and . (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.
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 , the radii increase in proportion to the square of the eigenvalue number ; see Meixner et al. (1980, §2.4). It is known that

where is the unique root of the equation in the interval , and . For and see §19.2(ii).

## §28.6(ii) Functions and

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

For ,

28.6.26

For the corresponding expansions of for change to 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 and ; compare Table 28.6.1 and (28.6.20).