§28.31 Equations of Whittaker–Hill and Ince

Keywords:: Ince’s equation, equation of Ince
Permalink:: http://dlmf.nist.gov/28.31

§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

Keywords:: Whittaker–Hill equation
Permalink:: http://dlmf.nist.gov/28.31.i

Hill’s equation with three terms

28.31.1 $W^{{\prime\prime}}+\left(A+B\mathop{\cos\/}\nolimits\!\left(2z\right)-\tfrac{1}{2}(kc)^{2}\mathop{\cos\/}\nolimits\!\left(4z\right)\right)W=0$

Symbols:: $\mathop{\cos\/}\nolimits z$ : 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
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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

Notes:: See Arscott (1967), Urwin and Arscott (1970).
Keywords:: Hill’s equation, Ince polynomials, equation of Ince
Permalink:: http://dlmf.nist.gov/28.31.ii

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

28.31.2

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

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

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

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : 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
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in (28.31.1). The result is the Equation of Ince:

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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
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Formal $2\pi$ -periodic solutions can be constructed as Fourier series; compare §28.4:

28.31.4 $w_{{\mathit{e},s}}(z)=\sum _{{\ell=0}}^{{\infty}}A_{{2\ell+s}}\mathop{\cos\/}\nolimits(2\ell+s)z,$ $s=0,1$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : 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

28.31.5 $w_{{\mathit{o},s}}(z)=\sum _{{\ell=0}}^{{\infty}}B_{{2\ell+s}}\mathop{\sin\/}\nolimits(2\ell+s)z,$ $s=1,2$ ,

Symbols:: $\mathop{\sin\/}\nolimits z$ : 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
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where the coefficients satisfy

28.31.6

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

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

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

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

28.31.7

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

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

Symbols:: $A$ : constant, $\eta$ : variable and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/28.31.E7
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28.31.8

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

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

Symbols:: $B$ : constant, $\eta$ : variable and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/28.31.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

28.31.9

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

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

Symbols:: $B$ : constant, $\eta$ : variable and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/28.31.E9
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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}$

Symbols:: $m$ : integer, $n$ : integer, $z$ : complex variable, $\xi$ : variable and $C_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Referenced by:: §28.31(iii)
Permalink:: http://dlmf.nist.gov/28.31.E10
Encodings:: TeX, pMML, png

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}$

Symbols:: $m$ : integer, $n$ : integer, $z$ : complex variable, $\xi$ : variable and $S_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Referenced by:: §28.31(iii)
Permalink:: http://dlmf.nist.gov/28.31.E11
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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 $\dfrac{1}{\pi}\int _{0}^{{2\pi}}\left(C_{p}^{m}(x,\xi)\right)^{2}dx=\dfrac{1}{\pi}\int _{0}^{{2\pi}}\left(S_{p}^{m}(x,\xi)\right)^{2}dx=1,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $m$ : integer, $x$ : real variable, $\xi$ : variable, $C_{{p}}^{{m}}(z,\xi)$ : Ince polynomials and $S_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Permalink:: http://dlmf.nist.gov/28.31.E12
Encodings:: TeX, pMML, png

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

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

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

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

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : integer, $x$ : real variable, $\xi$ : variable, $C_{{p}}^{{m}}(z,\xi)$ : Ince polynomials and $S_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Permalink:: http://dlmf.nist.gov/28.31.E13
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

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

$C_{p}^{m}(x,\xi)\to\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(x,q\right),$

$S_{p}^{m}(x,\xi)\to\mathop{\mathrm{se}_{{m}}\/}\nolimits\!\left(x,q\right),$

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, $q=h^{2}$ : parameter, $x$ : real variable, $\xi$ : variable, $C_{{p}}^{{m}}(z,\xi)$ : Ince polynomials and $S_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Permalink:: http://dlmf.nist.gov/28.31.E14
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28.31.15

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

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

Symbols:: $m$ : integer, $q=h^{2}$ : parameter and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/28.31.E15
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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

Notes:: See Urwin (1964, 1965).
Keywords:: paraboloidal wave functions, wave functions
Permalink:: http://dlmf.nist.gov/28.31.iii

With (28.31.10) and (28.31.11),

28.31.16 $\mathit{hc}_{p}^{m}(z,\xi)=e^{{-\frac{1}{4}\xi\mathop{\cos\/}\nolimits\!\left(2z\right)}}C_{p}^{m}(z,\xi),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $m$ : integer, $z$ : complex variable, $\mathit{hc}_{p}^{m}(z,\xi)$ : paraboloidal wave function and $C_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Permalink:: http://dlmf.nist.gov/28.31.E16
Encodings:: TeX, pMML, png

28.31.17 $\mathit{hs}_{p}^{m}(z,\xi)=e^{{-\frac{1}{4}\xi\mathop{\cos\/}\nolimits\!\left(2z\right)}}S_{p}^{m}(z,\xi),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $m$ : integer, $z$ : complex variable, $\mathit{hs}_{p}^{m}(z,\xi)$ : paraboloidal wave function and $S_{{p}}^{{m}}(z,\xi)$ : Ince polynomials
Permalink:: http://dlmf.nist.gov/28.31.E17
Encodings:: TeX, pMML, png

are called paraboloidal wave functions. They satisfy the differential equation

28.31.18 $w^{{\prime\prime}}+\left(\eta-\tfrac{1}{8}\xi^{2}-(p+1)\xi\mathop{\cos\/}\nolimits\!\left(2z\right)+\tfrac{1}{8}\xi^{2}\mathop{\cos\/}\nolimits\!\left(4z\right)\right)w=0,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : 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

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

For change of sign of $\xi$ ,

28.31.19

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

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

Symbols:: $m$ : 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

and

28.31.20

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

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

Symbols:: $m$ : 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

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

28.31.21 $\int _{0}^{{2\pi}}\mathit{hc}_{p}^{{m_{1}}}(x,\xi)\mathit{hc}_{p}^{{m_{2}}}(x,\xi)dx=\int _{0}^{{2\pi}}\mathit{hs}_{p}^{{m_{1}}}(x,\xi)\mathit{hs}_{p}^{{m_{2}}}(x,\xi)dx=0.$

Symbols:: $dx$ : 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

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 $\int _{{u_{0}}}^{{u_{{\infty}}}}\int _{0}^{{2\pi}}\mathit{hc}_{{p_{1}}}^{{m_{1}}}(u,\xi)\mathit{hc}_{{p_{1}}}^{{m_{1}}}(v,\xi)\mathit{hc}_{{p_{2}}}^{{m_{2}}}(u,\xi)\mathit{hc}_{{p_{2}}}^{{m_{2}}}(v,\xi)\*\left(\mathop{\cos\/}\nolimits\!\left(2u\right)-\mathop{\cos\/}\nolimits\!\left(2v\right)\right)dvdu=0,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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

and

28.31.23 $\int _{{u_{0}}}^{{u_{{\infty}}}}\int _{0}^{{2\pi}}\mathit{hs}_{{p_{1}}}^{{m_{1}}}(u,\xi)\mathit{hs}_{{p_{1}}}^{{m_{1}}}(v,\xi)\mathit{hs}_{{p_{2}}}^{{m_{2}}}(u,\xi)\mathit{hs}_{{p_{2}}}^{{m_{2}}}(v,\xi)\*\left(\mathop{\cos\/}\nolimits\!\left(2u\right)-\mathop{\cos\/}\nolimits\!\left(2v\right)\right)dvdu=0,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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

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

28.31.24 $\int _{{u_{0}}}^{{u_{{\infty}}}}\int _{0}^{{2\pi}}\mathit{hc}_{{p_{1}}}^{{m_{1}}}(u,\xi)\mathit{hc}_{{p_{1}}}^{{m_{1}}}(v,\xi)\mathit{hs}_{{p_{2}}}^{{m_{2}}}(u,\xi)\mathit{hs}_{{p_{2}}}^{{m_{2}}}(v,\xi)\*\left(\mathop{\cos\/}\nolimits\!\left(2u\right)-\mathop{\cos\/}\nolimits\!\left(2v\right)\right)dvdu=0,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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

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

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

¶ Asymptotic Behavior

Keywords:: paraboloidal wave functions

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

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

§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