§28.8 Asymptotic Expansions for Large $q$

Keywords:: Mathieu functions, Mathieu’s equation
Permalink:: http://dlmf.nist.gov/28.8

§28.8(i) Eigenvalues
§28.8(ii) Sips’ Expansions
§28.8(iii) Goldstein’s Expansions
§28.8(iv) Uniform Approximations

§28.8(i) Eigenvalues

Notes:: See Meixner and Schäfke (1954, pp. 134–139 and 210).
Referenced by:: §28.34(ii)
Permalink:: http://dlmf.nist.gov/28.8.i

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

28.8.1 $\rselection{\mathop{a_{{m}}\/}\nolimits\!\left(h^{2}\right)\\ \mathop{b_{{m+1}}\/}\nolimits\!\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h}(s^{3}+3s)-\frac{1}{2^{{12}}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{{17}}h^{3}}(33s^{5}+410s^{3}+405s)-\frac{1}{2^{{20}}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-\frac{1}{2^{{25}}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\sim$ : asymptotic equality, $m$ : 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

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $e$ : base of exponential function, $!$ : $n!$ : factorial, $m$ : integer and $h$ : parameter
A&S Ref:: 20.2.31
Permalink:: http://dlmf.nist.gov/28.8.E2
Encodings:: TeX, pMML, png

§28.8(ii) Sips’ Expansions

Keywords:: Mathieu functions
Referenced by:: §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.8.ii

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

$\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(x,h^{2}\right)=\widehat{C}_{m}\left(U_{m}(\xi)+V_{m}(\xi)\right),$

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

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : integer, $h$ : parameter, $x$ : real variable, $\xi$ : variable, $U_{m}(\xi)$ : function and $V_{m}(\xi)$ : function
A&S Ref:: 20.9.15 (in slightly different form) 20.9.16 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.8.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

where

28.8.4 $U_{m}(\xi)\sim\mathop{D_{{m}}\/}\nolimits\!\left(\xi\right)-\frac{1}{2^{6}h}\left(\mathop{D_{{m+4}}\/}\nolimits\!\left(\xi\right)-4!\dbinom{m}{4}\mathop{D_{{m-4}}\/}\nolimits\!\left(\xi\right)\right)+\frac{1}{2^{{13}}h^{2}}\left(\mathop{D_{{m+8}}\/}\nolimits\!\left(\xi\right)-2^{5}(m+2)\mathop{D_{{m+4}}\/}\nolimits\!\left(\xi\right)+4!\, 2^{5}(m-1)\dbinom{m}{4}\mathop{D_{{m-4}}\/}\nolimits\!\left(\xi\right)+8!\binom{m}{8}\mathop{D_{{m-8}}\/}\nolimits\!\left(\xi\right)\right)+\cdots,$

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 $V_{m}(\xi)\sim\frac{1}{2^{4}h}\bigg(-\mathop{D_{{m+2}}\/}\nolimits\!\left(\xi\right)-m(m-1)\mathop{D_{{m-2}}\/}\nolimits\!\left(\xi\right)\bigg)+\frac{1}{2^{{10}}h^{2}}\left(\mathop{D_{{m+6}}\/}\nolimits\!\left(\xi\right)+(m^{2}-25m-36)\mathop{D_{{m+2}}\/}\nolimits\!\left(\xi\right)-m(m-1)(m^{2}+27m-10)\mathop{D_{{m-2}}\/}\nolimits\!\left(\xi\right)+6!\binom{m}{6}\mathop{D_{{m-6}}\/}\nolimits\!\left(\xi\right)\right)+\cdots,$

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 $\widehat{C}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{{\ifrac{1}{4}}}\left(1+\frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right)^{{-\ifrac{1}{2}}},$

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 $\widehat{S}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{{\ifrac{1}{4}}}\left(1-\frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}-121m^{2}-122m-84}{2048h^{2}}+\cdots\right)^{{-\ifrac{1}{2}}}.$

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

Notes:: See Goldstein (1927).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.8.iii

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

$\dfrac{\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(x,h^{2}\right)}{\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(0,h^{2}\right)}=\dfrac{2^{{m-(\ifrac{1}{2})}}}{\sigma _{m}}\left(W_{m}^{{+}}(x)(P_{m}(x)-Q_{m}(x))+W_{m}^{{-}}(x)(P_{m}(x)+Q_{m}(x))\right),$

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

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : 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

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

$\sigma _{m}\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^{3}+59s}{2^{{11}}h^{3}}+\cdots,$

$\tau _{{m+1}}\sim 2h-\dfrac{1}{4}s-\dfrac{2s^{2}+3}{2^{6}h}-\frac{7s^{3}+47s}{2^{{10}}h^{2}}-\cdots,$

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 $P_{m}(x)\sim 1+\dfrac{s}{2^{3}h{\mathop{\cos\/}\nolimits^{{2}}}x}+\dfrac{1}{h^{2}}\left(\dfrac{s^{4}+86s^{2}+105}{2^{{11}}{\mathop{\cos\/}\nolimits^{{4}}}x}-\dfrac{s^{4}+22s^{2}+57}{2^{{11}}{\mathop{\cos\/}\nolimits^{{2}}}x}\right)+\cdots,$

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 $Q_{m}(x)\sim\dfrac{\mathop{\sin\/}\nolimits x}{{\mathop{\cos\/}\nolimits^{{2}}}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\mathop{\cos\/}\nolimits^{{2}}}x}\right)\right)+\cdots.$

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

§28.8(iv) Uniform Approximations

Referenced by:: §2.8(vi), §28.16, §28.26(ii), §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.8.iv

¶ Barrett’s Expansions

Keywords:: Mathieu functions, modified Mathieu functions

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

Keywords:: Mathieu functions

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).

§28.8 Asymptotic Expansions for Large

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