28 Mathieu Functions and Hill’s Equation Hill’s Equation 28.30 Expansions in Series of Eigenfunctions 28.32 Mathematical Applications

§28.31 Equations of Whittaker–Hill and Ince

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Keywords:: Ince’s equation, equation of Ince
Permalink:: http://dlmf.nist.gov/28.31
See also:: Annotations for Ch.28

§28.31(i) Whittaker–Hill Equation
§28.31(ii) Equation of Ince; Ince Polynomials
§28.31(iii) Paraboloidal Wave Functions

§28.31(i) Whittaker–Hill Equation

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Keywords:: Whittaker–Hill equation
Referenced by:: item (f)
Permalink:: http://dlmf.nist.gov/28.31.i
See also:: Annotations for §28.31 and Ch.28

Hill’s equation with three terms

28.31.1		$W^{\prime\prime}+\left(A+B\cos\left(2z\right)-\tfrac{1}{2}(kc)^{2}\cos\left(4z% \right)\right)W=0$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $W(z)$ : solution Referenced by: §28.31(ii), §28.32(ii) Permalink: http://dlmf.nist.gov/28.31.E1 Encodings: TeX, pMML, png See also: Annotations for §28.31(i), §28.31 and Ch.28

and constant values of $A, B, k$ , and $c$ , is called the Equation of Whittaker–Hill. It has been discussed in detail by Arscott (1967) for $k^{2}<0$ , and by Urwin and Arscott (1970) for $k^{2}>0$ .

§28.31(ii) Equation of Ince; Ince Polynomials

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Keywords:: Fourier-series solutions, Hill’s equation, Ince polynomials, equation of Ince, normalization, polynomial solutions, zeros
Notes:: See Arscott (1967), Urwin and Arscott (1970).
Permalink:: http://dlmf.nist.gov/28.31.ii
See also:: Annotations for §28.31 and Ch.28

When $k^{2}<0$ , we substitute

28.31.2	$\displaystyle\xi^{2}$	$\displaystyle=-4k^{2}c^{2},$
	$\displaystyle A$	$\displaystyle=\eta-\tfrac{1}{8}\xi^{2},$
	$\displaystyle B$	$\displaystyle=-(p+1)\xi,$
	$\displaystyle W(z)$	$\displaystyle=w(z)\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution, $W(z)$ : solution, $A$ : constant, $B$ : constant, $\eta$ : variable, $\xi$ : variable and $k$ : root Permalink: http://dlmf.nist.gov/28.31.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

in (28.31.1). The result is the Equation of Ince:

28.31.3		$w^{\prime\prime}+\xi\sin\left(2z\right)w^{\prime}+(\eta-p\xi\cos\left(2z\right% ))w=0.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution, $\eta$ : variable and $\xi$ : variable Referenced by: §28.31(ii) Permalink: http://dlmf.nist.gov/28.31.E3 Encodings: TeX, pMML, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

Formal $2\pi$ -periodic solutions can be constructed as Fourier series; compare §28.4:

28.31.4	$\displaystyle w_{\mathit{e},s}(z)$	$\displaystyle=\sum_{\ell=0}^{\infty}A_{2\ell+s}\cos(2\ell+s)z,$
$s=0,1$ ,
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $A$ : constant Referenced by: §28.31(ii) Permalink: http://dlmf.nist.gov/28.31.E4 Encodings: TeX, pMML, png See also: Annotations for §28.31(ii), §28.31 and Ch.28
28.31.5	$\displaystyle w_{\mathit{o},s}(z)$	$\displaystyle=\sum_{\ell=0}^{\infty}B_{2\ell+s}\sin(2\ell+s)z,$
$s=1,2$ ,
ⓘ Symbols: $\sin\NVar{z}$ : sine function, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $B$ : constant Referenced by: §28.31(ii) Permalink: http://dlmf.nist.gov/28.31.E5 Encodings: TeX, pMML, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

where the coefficients satisfy

28.31.6		$\displaystyle-2\eta A_{0}+(2+p)\xi A_{2}$	$\displaystyle=0,$
		$\displaystyle p\xi A_{0}+(4-\eta)A_{2}+\left(\tfrac{1}{2}p+2\right)\xi A_{4}$	$\displaystyle=0,$
		$\displaystyle(\tfrac{1}{2}p-\ell+1)\xi A_{2\ell-2}+\left(4\ell^{2}-\eta\right)% A_{2\ell}+(\tfrac{1}{2}p+\ell+1)\xi A_{2\ell+2}$	$\displaystyle=0,$
	$\ell\geq 2$ ,
ⓘ Symbols: $A$ : constant, $\eta$ : variable and $\xi$ : variable Permalink: http://dlmf.nist.gov/28.31.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

28.31.7		$\displaystyle\left(1-\eta+\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)A_{% 1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi A_{3}$	$\displaystyle=0,$
		$\displaystyle(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi A_{2\ell-1}+\left((2\ell+1)^% {2}-\eta\right)A_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi A_{2\ell+3}$	$\displaystyle=0,$
	$\ell\geq 1$ ,
ⓘ Symbols: $A$ : constant, $\eta$ : variable and $\xi$ : variable Permalink: http://dlmf.nist.gov/28.31.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

28.31.8		$\displaystyle\left(1-\eta-\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)B_{% 1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi B_{3}$	$\displaystyle=0,$
		$\displaystyle(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi B_{2\ell-1}+\left((2\ell+1)^% {2}-\eta\right)B_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi B_{2\ell+3}$	$\displaystyle=0,$
	$\ell\geq 1$ ,
ⓘ Symbols: $B$ : constant, $\eta$ : variable and $\xi$ : variable Permalink: http://dlmf.nist.gov/28.31.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

28.31.9		$\displaystyle(4-\eta)B_{2}+\left(\tfrac{1}{2}p+2\right)\xi B_{4}$	$\displaystyle=0,$
		$\displaystyle(\tfrac{1}{2}p-\ell+1)\xi B_{2\ell-2}+(4\ell^{2}-\eta)B_{2\ell}+(% \tfrac{1}{2}p+\ell+1)\xi B_{2\ell+2}$	$\displaystyle=0,$
	$\ell\geq 2$ .
ⓘ Symbols: $B$ : constant, $\eta$ : variable and $\xi$ : variable Permalink: http://dlmf.nist.gov/28.31.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

When $p$ is a nonnegative integer, the parameter $\eta$ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. They are denoted by

28.31.10		$\begin{array}[]{cl}C_{2n}^{2m}(z,\xi)&\mbox{with $p=2n$},\\ C_{2n+1}^{2m+1}(z,\xi)&\mbox{with $p=2n+1$},\end{array}$
ⓘ Defines: $C_{p}^{m}(z,\xi)$ : Ince polynomials (locally) Symbols: $m$ : integer, $n$ : integer, $z$ : complex variable and $\xi$ : variable Referenced by: §28.31(iii) Permalink: http://dlmf.nist.gov/28.31.E10 Encodings: TeX, pMML, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

28.31.11		$\begin{array}[]{cl}S_{2n+1}^{2m+1}(z,\xi)&\mbox{with $p=2n+1$},\\ S_{2n+2}^{2m+2}(z,\xi)&\mbox{with $p=2n+2$},\end{array}$
ⓘ Defines: $S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})$ : Ince polynomials (locally) Symbols: $m$ : integer, $n$ : integer, $x$ : real variable, $z$ : complex variable, $a$ : parameter and $\xi$ : variable Referenced by: §28.31(iii) Permalink: http://dlmf.nist.gov/28.31.E11 Encodings: TeX, pMML, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

and $m=0,1,\dots,n$ in all cases.

The values of $\eta$ corresponding to $C_{p}^{m}(z,\xi)$ , $S_{p}^{m}(z,\xi)$ are denoted by $a_{p}^{m}(\xi)$ , $b_{p}^{m}(\xi)$ , respectively. They are real and distinct, and can be ordered so that $C_{p}^{m}(z,\xi)$ and $S_{p}^{m}(z,\xi)$ have precisely $m$ zeros, all simple, in $0\leq z<\pi$ . The normalization is given by

28.31.12		$\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi)\right)^{2}\,\mathrm{d}x=% \dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)\right)^{2}\,\mathrm{d}x=1,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer, $x$ : real variable, $\xi$ : variable, $C_{p}^{m}(z,\xi)$ : Ince polynomials and $S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})$ : Ince polynomials Permalink: http://dlmf.nist.gov/28.31.E12 Encodings: TeX, pMML, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

ambiguities in sign being resolved by requiring $C_{p}^{m}(x,\xi)$ and ${S_{p}^{m}}^{\prime}(x,\xi)$ to be continuous functions of $x$ and positive when $x=0$ .

For $\xi\to 0$ , with $x$ fixed,

28.31.13		$\displaystyle C_{p}^{0}(x,\xi)$	$\displaystyle\to 1/{\sqrt{2}},$
		$\displaystyle C_{p}^{m}(x,\xi)$	$\displaystyle\to\cos\left(mx\right),$
		$\displaystyle S_{p}^{m}(x,\xi)$	$\displaystyle\to\sin\left(mx\right),$
	$m\neq 0$ ;
		$\displaystyle a_{p}^{m}(\xi),\,b_{p}^{m}(\xi)$	$\displaystyle\to m^{2}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $m$ : integer, $x$ : real variable, $\xi$ : variable, $C_{p}^{m}(z,\xi)$ : Ince polynomials and $S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})$ : Ince polynomials Permalink: http://dlmf.nist.gov/28.31.E13 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

If $p\to\infty$ and $\xi\to 0$ in such a way that $p\xi\to 2q$ , then in the notation of §§28.2(v) and 28.2(vi)

28.31.14	$\displaystyle C_{p}^{m}(x,\xi)$	$\displaystyle\to\operatorname{ce}_{m}\left(x,q\right),$
28.31.14	$\displaystyle S_{p}^{m}(x,\xi)$	$\displaystyle\to\operatorname{se}_{m}\left(x,q\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $m$ : integer, $q=h^{2}$ : parameter, $x$ : real variable, $\xi$ : variable, $C_{p}^{m}(z,\xi)$ : Ince polynomials and $S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})$ : Ince polynomials Permalink: http://dlmf.nist.gov/28.31.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

28.31.15	$\displaystyle a_{p}^{m}(\xi)$	$\displaystyle\to a_{m}(q),$
28.31.15	$\displaystyle b_{p}^{m}(\xi)$	$\displaystyle\to b_{m}(q).$
ⓘ Symbols: $m$ : integer, $q=h^{2}$ : parameter and $\xi$ : variable Permalink: http://dlmf.nist.gov/28.31.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(ii), §28.31 and Ch.28

For proofs and further information, including convergence of the series (28.31.4), (28.31.5), see Arscott (1967).

§28.31(iii) Paraboloidal Wave Functions

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Keywords:: orthogonality properties, paraboloidal, paraboloidal wave functions, reflection properties, wave functions
Notes:: See Urwin (1964, 1965).
Permalink:: http://dlmf.nist.gov/28.31.iii
See also:: Annotations for §28.31 and Ch.28

With (28.31.10) and (28.31.11),

28.31.16		$\mathit{hc}_{p}^{m}(z,\xi)=e^{-\frac{1}{4}\xi\cos\left(2z\right)}C_{p}^{m}(z,% \xi),$
ⓘ Defines: $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $m$ : integer, $z$ : complex variable and $C_{p}^{m}(z,\xi)$ : Ince polynomials Permalink: http://dlmf.nist.gov/28.31.E16 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

28.31.17		$\mathit{hs}_{p}^{m}(z,\xi)=e^{-\frac{1}{4}\xi\cos\left(2z\right)}S_{p}^{m}(z,% \xi),$
ⓘ Defines: $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $m$ : integer, $z$ : complex variable and $S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})$ : Ince polynomials Permalink: http://dlmf.nist.gov/28.31.E17 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

are called paraboloidal wave functions. They satisfy the differential equation

28.31.18		$w^{\prime\prime}+\left(\eta-\tfrac{1}{8}\xi^{2}-(p+1)\xi\cos\left(2z\right)+% \tfrac{1}{8}\xi^{2}\cos\left(4z\right)\right)w=0,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $\eta$ : change of variable Permalink: http://dlmf.nist.gov/28.31.E18 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

with $\eta=a_{p}^{m}(\xi)$ , $\eta=b_{p}^{m}(\xi)$ , respectively.

For change of sign of $\xi$ ,

28.31.19	$\displaystyle\mathit{hc}_{2n}^{2m}(z,-\xi)$	$\displaystyle=(-1)^{m}\mathit{hc}_{2n}^{2m}(\tfrac{1}{2}\pi-z,\xi),$
28.31.19	$\displaystyle\mathit{hc}_{2n+1}^{2m+1}(z,-\xi)$	$\displaystyle=(-1)^{m}\mathit{hs}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer, $z$ : complex variable, $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function and $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function Permalink: http://dlmf.nist.gov/28.31.E19 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

and

28.31.20	$\displaystyle\mathit{hs}_{2n+1}^{2m+1}(z,-\xi)$	$\displaystyle=(-1)^{m}\mathit{hc}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi),$
28.31.20	$\displaystyle\mathit{hs}_{2n+2}^{2m+2}(z,-\xi)$	$\displaystyle=(-1)^{m}\mathit{hs}_{2n+2}^{2m+2}(\tfrac{1}{2}\pi-z,\xi).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer, $z$ : complex variable, $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function and $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function Permalink: http://dlmf.nist.gov/28.31.E20 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

For $m_{1}\neq m_{2}$ ,

28.31.21		$\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)\mathit{hc}_{p}^{m_{2}}(x,\xi)\,% \mathrm{d}x=\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)\mathit{hs}_{p}^{m_{2% }}(x,\xi)\,\mathrm{d}x=0.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer, $x$ : real variable, $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function and $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function Permalink: http://dlmf.nist.gov/28.31.E21 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

More important are the double orthogonality relations for $p_{1}\neq p_{2}$ or $m_{1}\neq m_{2}$ or both, given by

28.31.22		$\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)% \mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hc% }_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)% \,\mathrm{d}v\,\mathrm{d}u=0,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function Permalink: http://dlmf.nist.gov/28.31.E22 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

and

28.31.23		$\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hs}_{p_{1}}^{m_{1}}(u,\xi)% \mathit{hs}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs% }_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)% \,\mathrm{d}v\,\mathrm{d}u=0,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function Permalink: http://dlmf.nist.gov/28.31.E23 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

and also for all $p_{1},p_{2},m_{1},m_{2}$ , given by

28.31.24		$\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)% \mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs% }_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)% \,\mathrm{d}v\,\mathrm{d}u=0,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer, $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function and $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function Permalink: http://dlmf.nist.gov/28.31.E24 Encodings: TeX, pMML, png See also: Annotations for §28.31(iii), §28.31 and Ch.28

where $(u_{0},u_{\infty})=(0,\mathrm{i}\infty)$ when $\xi>0$ , and $(u_{0},u_{\infty})=(\tfrac{1}{2}\pi,\tfrac{1}{2}\pi+\mathrm{i}\infty)$ when $\xi<0$ .

For proofs and further integral equations see Urwin (1964, 1965).

Asymptotic Behavior

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Keywords:: asymptotic behavior for large variable, paraboloidal wave functions
See also:: Annotations for §28.31(iii), §28.31 and Ch.28

For $\xi>0$ , the functions $\mathit{hc}_{p}^{m}(z,\xi)$ , $\mathit{hs}_{p}^{m}(z,\xi)$ behave asymptotically as multiples of $\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}$ as $z\to\pm\mathrm{i}\infty$ . All other periodic solutions behave as multiples of $\exp\left(\tfrac{1}{4}\xi\cos\left(2z\right)\right)(\cos z)^{-p-2}$ .

For $\xi>0$ , the functions $\mathit{hc}_{p}^{m}(z,-\xi)$ , $\mathit{hs}_{p}^{m}(z,-\xi)$ behave asymptotically as multiples of $\exp\left(\tfrac{1}{4}\xi\cos\left(2z\right)\right)(\cos z)^{-p-2}$ as $z\to\tfrac{1}{2}\pi\pm\mathrm{i}\infty$ . All other periodic solutions behave as multiples of $\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}$ .

28.30 Expansions in Series of Eigenfunctions 28.32 Mathematical Applications

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§28.31 Equations of Whittaker–Hill and Ince

Contents

§28.31(i) Whittaker–Hill Equation

§28.31(ii) Equation of Ince; Ince Polynomials

§28.31(iii) Paraboloidal Wave Functions

Asymptotic Behavior