# §28.31 Equations of Whittaker–Hill and Ince

## §28.31(i) Whittaker–Hill Equation

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 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

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

 28.31.2 $\displaystyle\xi^{2}$ $\displaystyle=-4k^{2}c^{2},$ $\displaystyle A$ $\displaystyle=\eta-\tfrac{1}{8}\xi^{2},$ $\displaystyle B$ $\displaystyle=-(p+1)\xi,$ $\displaystyle W(z)$ $\displaystyle=w(z)\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right),$

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

 28.31.3 $w^{\prime\prime}+\xi\sin\left(2z\right)w^{\prime}+(\eta-p\xi\cos\left(2z\right% ))w=0.$

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

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 $\displaystyle C_{p}^{0}(x,\xi)$ $\displaystyle\to 1/{\sqrt{2}},$ $\displaystyle C_{p}^{m}(x,\xi)$ $\displaystyle\to\cos\left(mx\right),$ $\displaystyle S_{p}^{m}(x,\xi)$ $\displaystyle\to\sin\left(mx\right),$ $m\neq 0$; $\displaystyle a_{p}^{m}(\xi),\,b_{p}^{m}(\xi)$ $\displaystyle\to m^{2}.$

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 $\displaystyle C_{p}^{m}(x,\xi)$ $\displaystyle\to\mathrm{ce}_{m}\left(x,q\right),$ $\displaystyle S_{p}^{m}(x,\xi)$ $\displaystyle\to\mathrm{se}_{m}\left(x,q\right),$
 28.31.15 $\displaystyle a_{p}^{m}(\xi)$ $\displaystyle\to a_{m}(q),$ $\displaystyle b_{p}^{m}(\xi)$ $\displaystyle\to b_{m}(q).$ ⓘ 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 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

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 exponential function, $m$: 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 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 exponential function, $m$: 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 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 28

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

For change of sign of $\xi$,

 28.31.19 $\displaystyle\mathit{hc}_{2n}^{2m}(z,-\xi)$ $\displaystyle=(-1)^{m}\mathit{hc}_{2n}^{2m}(\tfrac{1}{2}\pi-z,\xi),$ $\displaystyle\mathit{hc}_{2n+1}^{2m+1}(z,-\xi)$ $\displaystyle=(-1)^{m}\mathit{hs}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi),$

and

 28.31.20 $\displaystyle\mathit{hs}_{2n+1}^{2m+1}(z,-\xi)$ $\displaystyle=(-1)^{m}\mathit{hc}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi),$ $\displaystyle\mathit{hs}_{2n+2}^{2m+2}(z,-\xi)$ $\displaystyle=(-1)^{m}\mathit{hs}_{2n+2}^{2m+2}(\tfrac{1}{2}\pi-z,\xi).$

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

 28.31.21 $\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)\mathit{hc}_{p}^{m_{2}}(x,\xi)% \mathrm{d}x=\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)\mathit{hs}_{p}^{m_{2% }}(x,\xi)\mathrm{d}x=0.$

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 $\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)% \mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hc% }_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)% \mathrm{d}v\mathrm{d}u=0,$

and

 28.31.23 $\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hs}_{p_{1}}^{m_{1}}(u,\xi)% \mathit{hs}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs% }_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)% \mathrm{d}v\mathrm{d}u=0,$

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

 28.31.24 $\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)% \mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs% }_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)% \mathrm{d}v\mathrm{d}u=0,$

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

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 $\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}$.