28 Mathieu Functions and Hill’s Equation Mathieu Functions of Noninteger Order 28.15 Expansions for Small

§28.16 Asymptotic Expansions for Large

Notes:: See Meixner and Schäfke (1954, p. 139).
Keywords:: Mathieu’s equation
Referenced by:: §28.34(ii)
Permalink:: http://dlmf.nist.gov/28.16

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

28.16.1 $\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{% 1}{8}(s^{2}+1)-\dfrac{1}{2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{{12}}h^{2}}(5s^{4}+34s% ^{2}+9)-\dfrac{1}{2^{{17}}h^{3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{{20}}h^{4% }}(63s^{6}+1260s^{4}+2943s^{2}+486)-\dfrac{1}{2^{{25}}h^{5}}(527s^{7}+15617s^{% 5}+69001s^{3}+41607s)+\cdots.$

Symbols:: $\mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\sim$ : asymptotic equality, : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: TeX, pMML, png

For graphical interpretation, see Figures 28.13.1 and 28.13.2.