# §28.8 Asymptotic Expansions for Large

## §28.8(i) Eigenvalues

Denote and . Then as with ,

For error estimates see Kurz (1979), and for graphical interpretation see Figure 28.2.1. Also,

## §28.8(ii) Sips’ Expansions

Let , where is a real constant such that . Also let and 18.3). Then as

28.8.3

where

28.8.4
28.8.5

and

28.8.6
28.8.7

These results are derived formally in Sips (1949, 1959, 1965). See also Meixner and Schäfke (1954, §2.84).

## §28.8(iii) Goldstein’s Expansions

Let , where is a constant such that , and . Then as

28.8.8

where

28.8.9

and

28.8.10
28.8.11
28.8.12

## §28.8(iv) Uniform Approximations

### ¶ Barrett’s Expansions

Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). The approximations apply when the parameters and are real and large, and are uniform with respect to various regions in the -plane. The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included.

### ¶ Dunster’s Approximations

Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). These approximations apply when and are real and . They are uniform with respect to when , where is an arbitrary constant such that , and also with respect to in the semi-infinite strip given by and .

The approximations are expressed in terms of Whittaker functions and with ; compare §2.8(vi). They are derived by rigorous analysis and accompanied by strict and realistic error bounds. With additional restrictions on , uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii).

Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions 28.12(ii)) and modified Mathieu functions 28.20(iii)).

For related results see Langer (1934) and Sharples (1967, 1971).