# §28.8 Asymptotic Expansions for Large $q$

## §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 $\rselection{a_{m}\left(h^{2}\right)\\ b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h% }(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^% {5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-% \frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $m$: integer and $h$: parameter A&S Ref: 20.2.30 (in different form) Permalink: http://dlmf.nist.gov/28.8.E1 Encodings: TeX, pMML, png See also: Annotations for §28.8(i), §28.8 and Ch.28

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

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

## §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 $\xi=2\sqrt{h}\cos x$ and $D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)$18.3). Then as $h\to+\infty$

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

where

 28.8.4 $\displaystyle U_{m}(\xi)$ $\displaystyle\sim D_{m}\left(\xi\right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi% \right)-4!\dbinom{m}{4}D_{m-4}\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}% \left(D_{m+8}\left(\xi\right)-2^{5}(m+2)D_{m+4}\left(\xi\right)+4!\,2^{5}(m-1)% \dbinom{m}{4}D_{m-4}\left(\xi\right)+8!\genfrac{(}{)}{0.0pt}{}{m}{8}D_{m-8}% \left(\xi\right)\right)+\cdots,$ ⓘ Defines: $U_{m}(\xi)$: function (locally) Symbols: $D_{\NVar{\nu}}\left(\NVar{z}\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable A&S Ref: 20.9.17 (in different form) Permalink: http://dlmf.nist.gov/28.8.E4 Encodings: TeX, pMML, png See also: Annotations for §28.8(ii), §28.8 and Ch.28 28.8.5 $\displaystyle V_{m}(\xi)$ $\displaystyle\sim\frac{1}{2^{4}h}\bigg{(}-D_{m+2}\left(\xi\right)-m(m-1)D_{m-2% }\left(\xi\right)\bigg{)}+\frac{1}{2^{10}h^{2}}\left(D_{m+6}\left(\xi\right)+(% m^{2}-25m-36)D_{m+2}\left(\xi\right)-m(m-1)(m^{2}+27m-10)D_{m-2}\left(\xi% \right)-6!\genfrac{(}{)}{0.0pt}{}{m}{6}D_{m-6}\left(\xi\right)\right)+\cdots,$ ⓘ Defines: $V_{m}(\xi)$: function (locally) Symbols: $D_{\NVar{\nu}}\left(\NVar{z}\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable A&S Ref: 20.9.18 (in slightly different form) Referenced by: Erratum (V1.0.16) for Equation (28.8.5) Permalink: http://dlmf.nist.gov/28.8.E5 Encodings: TeX, pMML, png Error (effective with 1.0.16): Originally the $-$ in front of the $6!$ was given incorrectly as $+$. Suggested 2017-02-02 by Daniel Karlsson See also: Annotations for §28.8(ii), §28.8 and Ch.28

and

 28.8.6 $\displaystyle\widehat{C}_{m}$ $\displaystyle\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1+% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right% )^{-\ifrac{1}{2}},$ ⓘ Defines: $\widehat{C}_{m}$: coefficient (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: 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 See also: Annotations for §28.8(ii), §28.8 and Ch.28 28.8.7 $\displaystyle\widehat{S}_{m}$ $\displaystyle\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1-% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}-121m^{2}-122m-84}{2048h^{2}}+\cdots\right)% ^{-\ifrac{1}{2}}.$ ⓘ Defines: $\widehat{S}_{m}$: coefficient (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: 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 See also: Annotations for §28.8(ii), §28.8 and Ch.28

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 $\displaystyle\dfrac{\mathrm{ce}_{m}\left(x,h^{2}\right)}{\mathrm{ce}_{m}\left(% 0,h^{2}\right)}$ $\displaystyle=\dfrac{2^{m-(\ifrac{1}{2})}}{\sigma_{m}}\left(W_{m}^{+}(x)(P_{m}% (x)-Q_{m}(x))+W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right),$ $\displaystyle\dfrac{\mathrm{se}_{m+1}\left(x,h^{2}\right)}{\mathrm{se}_{m+1}'% \left(0,h^{2}\right)}$ $\displaystyle=\dfrac{2^{m-(\ifrac{1}{2})}}{\tau_{m+1}}\left(W_{m}^{+}(x)(P_{m}% (x)-Q_{m}(x))-W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right),$ ⓘ Symbols: $\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $m$: integer, $h$: parameter, $x$: real variable, $Q_{m}(x)$, $W_{m}^{\pm}$: function, $\sigma_{m}$, $\tau_{m+1}$ and $P_{m}(x)$ A&S Ref: 20.9.11 (in slightly different form) 20.9.12 (in slightly different form) Permalink: http://dlmf.nist.gov/28.8.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.8(iii), §28.8 and Ch.28

where

 28.8.9 $W_{m}^{\pm}(x)=\frac{e^{\pm 2h\sin x}}{(\cos x)^{m+1}}\begin{cases}\left(\cos% \left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\\ \left(\sin\left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\end{cases}$ ⓘ Defines: $W_{m}^{\pm}$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin\NVar{z}$: sine function, $m$: integer, $h$: parameter and $x$: real variable A&S Ref: 20.9.13 (in slightly different form) Permalink: http://dlmf.nist.gov/28.8.E9 Encodings: TeX, pMML, png See also: Annotations for §28.8(iii), §28.8 and Ch.28

and

 28.8.10 $\displaystyle\sigma_{m}$ $\displaystyle\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^% {3}+59s}{2^{11}h^{3}}+\cdots,$ $\displaystyle\tau_{m+1}$ $\displaystyle\sim 2h-\dfrac{1}{4}s-\dfrac{2s^{2}+3}{2^{6}h}-\frac{7s^{3}+47s}{% 2^{10}h^{2}}-\cdots,$ ⓘ Defines: $\sigma_{m}$ (locally) and $\tau_{m+1}$ (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $m$: integer and $h$: parameter A&S Ref: 20.9.14 (in different form) Permalink: http://dlmf.nist.gov/28.8.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.8(iii), §28.8 and Ch.28
 28.8.11 $\displaystyle P_{m}(x)$ $\displaystyle\sim 1+\dfrac{s}{2^{3}h{\cos}^{2}x}+\dfrac{1}{h^{2}}\left(\dfrac{% s^{4}+86s^{2}+105}{2^{11}{\cos}^{4}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos}^{2% }x}\right)+\cdots,$ ⓘ Defines: $P_{m}(x)$ (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $\cos\NVar{z}$: cosine function, $m$: integer, $h$: parameter and $x$: real variable A&S Ref: 20.9.14 (in different form) Permalink: http://dlmf.nist.gov/28.8.E11 Encodings: TeX, pMML, png See also: Annotations for §28.8(iii), §28.8 and Ch.28 28.8.12 $\displaystyle Q_{m}(x)$ $\displaystyle\sim\dfrac{\sin x}{{\cos}^{2}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+% \dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cos}^{2}x}\right)% \right)+\cdots.$ ⓘ Defines: $Q_{m}(x)$ (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $m$: integer, $h$: parameter and $x$: real variable A&S Ref: 20.9.14 (in different form) Permalink: http://dlmf.nist.gov/28.8.E12 Encodings: TeX, pMML, png See also: Annotations for §28.8(iii), §28.8 and Ch.28

## §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 $a$ 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 $0\leq\Re z\leq\pi$ and $\Im z\geq 0$.

The approximations are expressed in terms of Whittaker functions $W_{\kappa,\mu}\left(z\right)$ and $M_{\kappa,\mu}\left(z\right)$ 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 $\mathrm{me}_{\nu}\left(z,q\right)$28.12(ii)) and modified Mathieu functions ${\mathrm{M}^{(j)}_{\nu}}\left(z,h\right)$28.20(iii)).

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