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 $W^{\prime\prime}+\left(A+B\cos\left(2z\right)-\tfrac{1}{2}(kc)^{2}\cos\left(4z% \right)\right)W=0$

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Symbols:: $\cos\NVar{z}$ : cosine function, : complex variable and : 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 , and , 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

$\xi^{2}=-4k^{2}c^{2},$

$A=\eta-\tfrac{1}{8}\xi^{2},$

$B=-(p+1)\xi,$

$W(z)=w(z)\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right),$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, : complex variable, : Mathieu’s equation solution, : solution, : constant, : constant, $\eta$ : variable, $\xi$ : variable and : 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 $w^{\prime\prime}+\xi\sin\left(2z\right)w^{\prime}+(\eta-p\xi\cos\left(2z\right% ))w=0.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : complex variable, : 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 $w_{\mathit{e},s}(z)=\sum_{\ell=0}^{\infty}A_{2\ell+s}\cos(2\ell+s)z,$

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Symbols:: $\cos\NVar{z}$ : cosine function, : complex variable, : Mathieu’s equation solution and : 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 $w_{\mathit{o},s}(z)=\sum_{\ell=0}^{\infty}B_{2\ell+s}\sin(2\ell+s)z,$

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Symbols:: $\sin\NVar{z}$ : sine function, : complex variable, : Mathieu’s equation solution and : 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

$-2\eta A_{0}+(2+p)\xi A_{2}=0,$

$p\xi A_{0}+(4-\eta)A_{2}+\left(\tfrac{1}{2}p+2\right)\xi A_{4}=0,$

$(\tfrac{1}{2}p-\ell+1)\xi A_{2\ell-2}+\left(4\ell^{2}-\eta\right)A_{2\ell}+(% \tfrac{1}{2}p+\ell+1)\xi A_{2\ell+2}=0,$ $\ell\geq 2$ ,

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Symbols:: : 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

$\left(1-\eta+\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)A_{1}+\left(% \tfrac{1}{2}p+\tfrac{3}{2}\right)\xi A_{3}=0,$

$(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi A_{2\ell-1}+\left((2\ell+1)^{2}-\eta% \right)A_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi A_{2\ell+3}=0,$ $\ell\geq 1$ ,

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Symbols:: : 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

$\left(1-\eta-\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)B_{1}+\left(% \tfrac{1}{2}p+\tfrac{3}{2}\right)\xi B_{3}=0,$

$(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi B_{2\ell-1}+\left((2\ell+1)^{2}-\eta% \right)B_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi B_{2\ell+3}=0,$ $\ell\geq 1$ ,

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Symbols:: : 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

$(4-\eta)B_{2}+\left(\tfrac{1}{2}p+2\right)\xi B_{4}=0,$

$(\tfrac{1}{2}p-\ell+1)\xi B_{2\ell-2}+(4\ell^{2}-\eta)B_{2\ell}+(\tfrac{1}{2}p% +\ell+1)\xi B_{2\ell+2}=0,$ $\ell\geq 2$ .

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Symbols:: : 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 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}$

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Defines:: $C_{p}^{m}(z,\xi)$ : Ince polynomials (locally)
Symbols:: : integer, : integer, : 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}$

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Defines:: $S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})$ : Ince polynomials (locally)
Symbols:: : integer, : integer, : real variable, : complex variable, : 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 zeros, all simple, in $0\leq z<\pi$ . The normalization is given by

28.31.12 $\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi)\right)^{2}\,\mathrm{d}x=% \dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)\right)^{2}\,\mathrm{d}x=1,$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, : 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 and positive when .

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

28.31.13

$C_{p}^{0}(x,\xi)\to 1/{\sqrt{2}},$

$C_{p}^{m}(x,\xi)\to\cos\left(mx\right),$

$S_{p}^{m}(x,\xi)\to\sin\left(mx\right),$ $m\neq 0$ ;

$a_{p}^{m}(\xi),\,b_{p}^{m}(\xi)\to m^{2}.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : integer, : 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

$C_{p}^{m}(x,\xi)\to\operatorname{ce}_{m}\left(x,q\right),$

$S_{p}^{m}(x,\xi)\to\operatorname{se}_{m}\left(x,q\right),$

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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, : integer, $q=h^{2}$ : parameter, : 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

$a_{p}^{m}(\xi)\to a_{m}(q),$

$b_{p}^{m}(\xi)\to b_{m}(q).$

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Symbols:: : 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 $\mathit{hc}_{p}^{m}(z,\xi)=e^{-\frac{1}{4}\xi\cos\left(2z\right)}C_{p}^{m}(z,% \xi),$

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Defines:: $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function (locally)
Symbols:: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, : integer, : 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 $\mathit{hs}_{p}^{m}(z,\xi)=e^{-\frac{1}{4}\xi\cos\left(2z\right)}S_{p}^{m}(z,% \xi),$

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Defines:: $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function (locally)
Symbols:: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, : integer, : 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 $w^{\prime\prime}+\left(\eta-\tfrac{1}{8}\xi^{2}-(p+1)\xi\cos\left(2z\right)+% \tfrac{1}{8}\xi^{2}\cos\left(4z\right)\right)w=0,$

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Symbols:: $\cos\NVar{z}$ : cosine function, : complex variable, : 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

$\mathit{hc}_{2n}^{2m}(z,-\xi)=(-1)^{m}\mathit{hc}_{2n}^{2m}(\tfrac{1}{2}\pi-z,% \xi),$

$\mathit{hc}_{2n+1}^{2m+1}(z,-\xi)=(-1)^{m}\mathit{hs}_{2n+1}^{2m+1}(\tfrac{1}{% 2}\pi-z,\xi),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, : integer, : 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

$\mathit{hs}_{2n+1}^{2m+1}(z,-\xi)=(-1)^{m}\mathit{hc}_{2n+1}^{2m+1}(\tfrac{1}{% 2}\pi-z,\xi),$

$\mathit{hs}_{2n+2}^{2m+2}(z,-\xi)=(-1)^{m}\mathit{hs}_{2n+2}^{2m+2}(\tfrac{1}{% 2}\pi-z,\xi).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, : integer, : 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 $\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)\mathit{hc}_{p}^{m_{2}}(x,\xi)\,% \mathrm{d}x=\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)\mathit{hs}_{p}^{m_{2% }}(x,\xi)\,\mathrm{d}x=0.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, : 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 , $\int$ : integral, : 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 , $\int$ : integral, : 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 , $\int$ : integral, : 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