q

x\to\pm\infty

§28.16 Asymptotic Expansions for Large $q$

ⓘ

Keywords:: Mathieu’s equation, asymptotic expansions for large $q$ , eigenvalues (or characteristic values)
Notes:: See Meixner and Schäfke (1954, p. 139).
Referenced by:: item (b)
Permalink:: http://dlmf.nist.gov/28.16
See also:: Annotations for Ch.28

Let $s=2m+1$ , $m=0,1,2,\dots$ , and $\nu$ be fixed with $m<\nu<m+1$ . Then as $h(=\sqrt{q})\to+\infty$

28.16.1		$\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$
ⓘ Symbols: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/28.16.E1 Encodings: TeX, pMML, png See also: Annotations for §28.16 and Ch.28

For graphical interpretation, see Figures 28.13.1 and 28.13.2.

§28.16 Asymptotic Expansions for Large q