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28 Mathieu Functions and Hill’s EquationHill’s Equation

§28.31 Equations of Whittaker–Hill and Ince

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§28.31(i) Whittaker–Hill Equation

Hill’s equation with three terms

28.31.1W^{{\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

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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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

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

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,
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,
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,
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.

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}
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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}
Show Annotations
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
Encodings:
TeX, pMML, png

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

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.

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)

28.31.15
a_{p}^{m}(\xi)\to a_{m}(q),
b_{p}^{m}(\xi)\to b_{m}(q).

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\mathop{\cos\/}\nolimits\!\left(% 2z\right)}}C_{p}^{m}(z,\xi),
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),

are called paraboloidal wave functions. They satisfy the differential equation

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

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

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

and

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

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

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}.

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