§28.8(i) Eigenvalues

Denote $h=\sqrt{q}$ and $s=2m+1$. Then as $h\to+\infty$ with $m=0,1,2,\dots$,

 28.8.1 $\rselection{\mathop{a_{m}\/}\nolimits\!\left(h^{2}\right)\\ \mathop{b_{m+1}\/}\nolimits\!\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^% {2}+1)-\frac{1}{2^{7}h}(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-% \frac{1}{2^{17}h^{3}}(33s^{5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+126% 0s^{4}+2943s^{2}+486)-\frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+416% 07s)+\cdots.$

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

 28.8.2 $\mathop{b_{m+1}\/}\nolimits\!\left(h^{2}\right)-\mathop{a_{m}\/}\nolimits\!% \left(h^{2}\right)=\frac{2^{4m+5}}{m!}\left(\frac{2}{\pi}\right)^{\ifrac{1}{2}% }h^{m+(\ifrac{3}{2})}e^{-4h}\*{\left(1-\frac{6m^{2}+14m+7}{32h}+\mathop{O\/}% \nolimits\!\left(\frac{1}{h^{2}}\right)\right)}.$

§28.8(ii) Sips’ Expansions

Let $x=\tfrac{1}{2}\pi+\lambda h^{-\ifrac{1}{4}}$, where $\lambda$ is a real constant such that $|\lambda|<2^{\ifrac{1}{4}}$. Also let $\xi=2\sqrt{h}\mathop{\cos\/}\nolimits x$ and $\mathop{D_{m}\/}\nolimits\!\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathop{% \mathit{He}_{m}\/}\nolimits\!\left(\xi\right)$18.3). Then as $h\to+\infty$

 28.8.3 $\displaystyle\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(x,h^{2}\right)$ $\displaystyle=\widehat{C}_{m}\left(U_{m}(\xi)+V_{m}(\xi)\right),$ $\displaystyle\frac{\mathop{\mathrm{se}_{m+1}\/}\nolimits\!\left(x,h^{2}\right)% }{\mathop{\sin\/}\nolimits x}$ $\displaystyle=\widehat{S}_{m}\left(U_{m}(\xi)-V_{m}(\xi)\right),$

where

 28.8.4 $\displaystyle U_{m}(\xi)$ $\displaystyle\sim\mathop{D_{m}\/}\nolimits\!\left(\xi\right)-\frac{1}{2^{6}h}% \left(\mathop{D_{m+4}\/}\nolimits\!\left(\xi\right)-4!\dbinom{m}{4}\mathop{D_{% m-4}\/}\nolimits\!\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}\left(\mathop{D% _{m+8}\/}\nolimits\!\left(\xi\right)-2^{5}(m+2)\mathop{D_{m+4}\/}\nolimits\!% \left(\xi\right)+4!\,2^{5}(m-1)\dbinom{m}{4}\mathop{D_{m-4}\/}\nolimits\!\left% (\xi\right)+8!\binom{m}{8}\mathop{D_{m-8}\/}\nolimits\!\left(\xi\right)\right)% +\cdots,$ Symbols: $\mathop{D_{\nu}\/}\nolimits\!\left(z\right)$: parabolic cylinder function, $\binom{m}{n}$: binomial coefficient, $!$: $n!$: factorial, $\sim$: asymptotic equality, $m$: integer, $h$: parameter, $\xi$: variable and $U_{m}(\xi)$: function A&S Ref: 20.9.17 (in different form) Permalink: http://dlmf.nist.gov/28.8.E4 Encodings: TeX, pMML, png 28.8.5 $\displaystyle V_{m}(\xi)$ $\displaystyle\sim\frac{1}{2^{4}h}\bigg(-\mathop{D_{m+2}\/}\nolimits\!\left(\xi% \right)-m(m-1)\mathop{D_{m-2}\/}\nolimits\!\left(\xi\right)\bigg)+\frac{1}{2^{% 10}h^{2}}\left(\mathop{D_{m+6}\/}\nolimits\!\left(\xi\right)+(m^{2}-25m-36)% \mathop{D_{m+2}\/}\nolimits\!\left(\xi\right)-m(m-1)(m^{2}+27m-10)\mathop{D_{m% -2}\/}\nolimits\!\left(\xi\right)+6!\binom{m}{6}\mathop{D_{m-6}\/}\nolimits\!% \left(\xi\right)\right)+\cdots,$ Symbols: $\mathop{D_{\nu}\/}\nolimits\!\left(z\right)$: parabolic cylinder function, $\binom{m}{n}$: binomial coefficient, $!$: $n!$: factorial, $\sim$: asymptotic equality, $m$: integer, $h$: parameter, $\xi$: variable and $V_{m}(\xi)$: function A&S Ref: 20.9.18 (in slightly different form) Permalink: http://dlmf.nist.gov/28.8.E5 Encodings: TeX, pMML, png

and

 28.8.6 $\displaystyle\widehat{C}_{m}$ $\displaystyle\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1+% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right% )^{-\ifrac{1}{2}},$ Symbols: $!$: $n!$: factorial, $\sim$: asymptotic equality, $m$: integer and $h$: parameter A&S Ref: 20.9.19 (in slightly different form) Permalink: http://dlmf.nist.gov/28.8.E6 Encodings: TeX, pMML, png 28.8.7 $\displaystyle\widehat{S}_{m}$ $\displaystyle\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1-% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}-121m^{2}-122m-84}{2048h^{2}}+\cdots\right)% ^{-\ifrac{1}{2}}.$ Symbols: $!$: $n!$: factorial, $\sim$: asymptotic equality, $m$: integer and $h$: parameter A&S Ref: 20.9.20 (in slightly different form) Permalink: http://dlmf.nist.gov/28.8.E7 Encodings: TeX, pMML, png

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 $x=\tfrac{1}{2}\pi-\mu h^{-\ifrac{1}{4}}$, where $\mu$ is a constant such that $\mu\geq 1$, and $s=2m+1$. Then as $h\to+\infty$

 28.8.8 $\displaystyle\dfrac{\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(x,h^{2}\right)}% {\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(0,h^{2}\right)}$ $\displaystyle=\dfrac{2^{m-(\ifrac{1}{2})}}{\sigma_{m}}\left(W_{m}^{+}(x)(P_{m}% (x)-Q_{m}(x))+W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right),$ $\displaystyle\dfrac{\mathop{\mathrm{se}_{m+1}\/}\nolimits\!\left(x,h^{2}\right% )}{{\mathop{\mathrm{se}_{m+1}\/}\nolimits^{\prime}}\!\left(0,h^{2}\right)}$ $\displaystyle=\dfrac{2^{m-(\ifrac{1}{2})}}{\tau_{m+1}}\left(W_{m}^{+}(x)(P_{m}% (x)-Q_{m}(x))-W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right),$

where

 28.8.9 $W_{m}^{\pm}(x)=\frac{e^{\pm 2h\mathop{\sin\/}\nolimits x}}{(\mathop{\cos\/}% \nolimits x)^{m+1}}\begin{cases}\left(\mathop{\cos\/}\nolimits\!\left(\frac{1}% {2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\\ \left(\mathop{\sin\/}\nolimits\!\left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right% )^{2m+1},\end{cases}$ Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $e$: base of exponential function, $\mathop{\sin\/}\nolimits z$: sine function, $m$: integer, $h$: parameter, $x$: real variable and $W_{m}^{\pm}$: function A&S Ref: 20.9.13 (in slightly different form) Permalink: http://dlmf.nist.gov/28.8.E9 Encodings: TeX, pMML, png

and

 28.8.10 $\displaystyle\sigma_{m}$ $\displaystyle\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^% {3}+59s}{2^{11}h^{3}}+\cdots,$ $\displaystyle\tau_{m+1}$ $\displaystyle\sim 2h-\dfrac{1}{4}s-\dfrac{2s^{2}+3}{2^{6}h}-\frac{7s^{3}+47s}{% 2^{10}h^{2}}-\cdots,$ Symbols: $\sim$: asymptotic equality, $m$: integer, $h$: parameter, $\sigma_{m}$ and $\tau_{m+1}$ A&S Ref: 20.9.14 (in different form) Permalink: http://dlmf.nist.gov/28.8.E10 Encodings: TeX, TeX, pMML, pMML, png, png
 28.8.11 $\displaystyle P_{m}(x)$ $\displaystyle\sim 1+\dfrac{s}{2^{3}h{\mathop{\cos\/}\nolimits^{2}}x}+\dfrac{1}% {h^{2}}\left(\dfrac{s^{4}+86s^{2}+105}{2^{11}{\mathop{\cos\/}\nolimits^{4}}x}-% \dfrac{s^{4}+22s^{2}+57}{2^{11}{\mathop{\cos\/}\nolimits^{2}}x}\right)+\cdots,$ Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $\sim$: asymptotic equality, $m$: integer, $h$: parameter, $x$: real variable and $P_{m}(x)$ A&S Ref: 20.9.14 (in different form) Permalink: http://dlmf.nist.gov/28.8.E11 Encodings: TeX, pMML, png 28.8.12 $\displaystyle Q_{m}(x)$ $\displaystyle\sim\dfrac{\mathop{\sin\/}\nolimits x}{{\mathop{\cos\/}\nolimits^% {2}}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+% \dfrac{4s^{3}+44s}{{\mathop{\cos\/}\nolimits^{2}}x}\right)\right)+\cdots.$ Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $\sim$: asymptotic equality, $\mathop{\sin\/}\nolimits z$: sine function, $m$: integer, $h$: parameter, $x$: real variable and $Q_{m}(x)$ A&S Ref: 20.9.14 (in different form) Permalink: http://dlmf.nist.gov/28.8.E12 Encodings: TeX, pMML, png

¶ 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 $a$ and $q$ are real and large, and are uniform with respect to various regions in the $z$-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 $q$ and $a$ are real and $q\to\infty$. They are uniform with respect to $a$ when $-2q\leq a\leq(2-\delta)q$, where $\delta$ is an arbitrary constant such that $0<\delta<4$, and also with respect to $z$ in the semi-infinite strip given by $0\leq\realpart{z}\leq\pi$ and $\imagpart{z}\geq 0$.

The approximations are expressed in terms of Whittaker functions $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ and $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ with $\mu=\tfrac{1}{4}$; compare §2.8(vi). They are derived by rigorous analysis and accompanied by strict and realistic error bounds. With additional restrictions on $z$, 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 $\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,q\right)$28.12(ii)) and modified Mathieu functions $\mathop{{\mathrm{M}^{(j)}_{\nu}}\/}\nolimits\!\left(z,h\right)$28.20(iii)).

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