§28.31 Equations of Whittaker–Hill and Ince

Hill’s equation with three terms

28.31.1		$$W^{\prime \prime }+\left(A+B\cos \left(2z\right)-\frac{1}{2}{(kc)}^2\cos \left(4z\right)\right)W=0$$
ⓘ Symbols: $\cos z$: 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 , and by Urwin and Arscott (1970) for $k^2>0$.

When , we substitute

28.31.2		${\xi }^2$	$=-4k^2{c}^2,$
		$A$	$=\eta -\frac{1}{8}{\xi }^2,$
		$B$	$=-(p+1)\xi ,$
		$W(z)$	$=w(z)\exp \left(-\frac{1}{4}\xi \cos \left(2z\right)\right),$
ⓘ Symbols: $\cos z$: cosine function, $\exp z$: 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: $\cos z$: cosine function, $\sin z$: 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		$w_{e,s}(z)$	$={\displaystyle \sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}}{A}_{2\mathrm{\ell }+s}\cos (2\mathrm{\ell }+s)z,$
$s=0,1$,
ⓘ Symbols: $\cos z$: 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		$w_{o,s}(z)$	$={\displaystyle \sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}}{B}_{2\mathrm{\ell }+s}\sin (2\mathrm{\ell }+s)z,$
$s=1,2$,
ⓘ Symbols: $\sin z$: 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		$-2\eta A_0+(2+p)\xi A_2$	$=0,$
		$p\xi A_0+(4-\eta )A_2+\left(\frac{1}{2}p+2\right)\xi A_4$	$=0,$
		$(\frac{1}{2}p-\mathrm{\ell }+1)\xi A_{2\mathrm{\ell }-2}+\left(4{\mathrm{\ell }}^2-\eta \right)A_{2\mathrm{\ell }}+(\frac{1}{2}p+\mathrm{\ell }+1)\xi A_{2\mathrm{\ell }+2}$	$=0,$
	$\mathrm{\ell }\ge 2$,
ⓘ 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		$\left(1-\eta +\left(\frac{1}{2}p+\frac{1}{2}\right)\xi \right)A_1+\left(\frac{1}{2}p+\frac{3}{2}\right)\xi A_3$	$=0,$
		$(\frac{1}{2}p-\mathrm{\ell }+\frac{1}{2})\xi A_{2\mathrm{\ell }-1}+\left({(2\mathrm{\ell }+1)}^2-\eta \right)A_{2\mathrm{\ell }+1}+(\frac{1}{2}p+\mathrm{\ell }+\frac{3}{2})\xi A_{2\mathrm{\ell }+3}$	$=0,$
	$\mathrm{\ell }\ge 1$,
ⓘ 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		$\left(1-\eta -\left(\frac{1}{2}p+\frac{1}{2}\right)\xi \right)B_1+\left(\frac{1}{2}p+\frac{3}{2}\right)\xi B_3$	$=0,$
		$(\frac{1}{2}p-\mathrm{\ell }+\frac{1}{2})\xi B_{2\mathrm{\ell }-1}+\left({(2\mathrm{\ell }+1)}^2-\eta \right)B_{2\mathrm{\ell }+1}+(\frac{1}{2}p+\mathrm{\ell }+\frac{3}{2})\xi B_{2\mathrm{\ell }+3}$	$=0,$
	$\mathrm{\ell }\ge 1$,
ⓘ 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		$(4-\eta )B_2+\left(\frac{1}{2}p+2\right)\xi B_4$	$=0,$
		$(\frac{1}{2}p-\mathrm{\ell }+1)\xi B_{2\mathrm{\ell }-2}+(4{\mathrm{\ell }}^2-\eta )B_{2\mathrm{\ell }}+(\frac{1}{2}p+\mathrm{\ell }+1)\xi B_{2\mathrm{\ell }+2}$	$=0,$
	$\mathrm{\ell }\ge 2$.
ⓘ 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
ⓘ 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
ⓘ Defines: $S_p^m(z,NVarxi)$: 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 $m=0,1,\mathrm{\dots },n$ 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 . The normalization is given by

28.31.12		$$\frac{1}{\pi }{\int }_0^{2\pi }{\left(C_p^m(x,\xi )\right)}^{2}dx=\frac{1}{\pi }{\int }_0^{2\pi }{\left(S_p^m(x,\xi )\right)}^{2}dx=1,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $m$: integer, $x$: real variable, $\xi$: variable, $C_p^m(z,\xi )$: Ince polynomials and $S_p^m(z,NVarxi)$: 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		$C_p^0(x,\xi )$	$\to 1/\sqrt{2},$
		$C_p^m(x,\xi )$	$\to \cos \left(mx\right),$
		$S_p^m(x,\xi )$	$\to \sin \left(mx\right),$
	$m\ne 0$;
		$a_p^m(\xi ),b_p^m(\xi )$	$\to m^2.$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $x$: real variable, $\xi$: variable, $C_p^m(z,\xi )$: Ince polynomials and $S_p^m(z,NVarxi)$: 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 $p\to \mathrm{\infty }$ 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		$C_p^m(x,\xi )$	$\to {\mathrm{ce}}_m(x,q),$
		$S_p^m(x,\xi )$	$\to {\mathrm{se}}_m(x,q),$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $m$: integer, $q=h^2$: parameter, $x$: real variable, $\xi$: variable, $C_p^m(z,\xi )$: Ince polynomials and $S_p^m(z,NVarxi)$: 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		$a_p^m(\xi )$	$\to a_m(q),$
		$b_p^m(\xi )$	$\to b_m(q).$
ⓘ 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).

With (28.31.10) and (28.31.11),

28.31.16		$${\mathit{hc}}_p^m(z,\xi )={\mathrm{e}}^{-\frac{1}{4}\xi \cos \left(2z\right)}{C}_p^m(z,\xi ),$$
ⓘ Defines: ${\mathit{hc}}_p^m(z,\xi )$: paraboloidal wave function (locally) Symbols: $\cos z$: 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		$${\mathit{hs}}_p^m(z,\xi )={\mathrm{e}}^{-\frac{1}{4}\xi \cos \left(2z\right)}{S}_p^m(z,\xi ),$$
ⓘ Defines: ${\mathit{hs}}_p^m(z,\xi )$: paraboloidal wave function (locally) Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $m$: integer, $z$: complex variable and $S_p^m(z,NVarxi)$: 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		$$w^{\prime \prime }+\left(\eta -\frac{1}{8}{\xi }^2-(p+1)\xi \cos \left(2z\right)+\frac{1}{8}{\xi }^2\cos \left(4z\right)\right)w=0,$$
ⓘ Symbols: $\cos z$: 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		${\mathit{hc}}_{2n}^{2m}(z,-\xi )$	$={(-1)}^m{\mathit{hc}}_{2n}^{2m}(\frac{1}{2}\pi -z,\xi ),$
		${\mathit{hc}}_{2n+1}^{2m+1}(z,-\xi )$	$={(-1)}^m{\mathit{hs}}_{2n+1}^{2m+1}(\frac{1}{2}\pi -z,\xi ),$
ⓘ 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		${\mathit{hs}}_{2n+1}^{2m+1}(z,-\xi )$	$={(-1)}^m{\mathit{hc}}_{2n+1}^{2m+1}(\frac{1}{2}\pi -z,\xi ),$
		${\mathit{hs}}_{2n+2}^{2m+2}(z,-\xi )$	$={(-1)}^m{\mathit{hs}}_{2n+2}^{2m+2}(\frac{1}{2}\pi -z,\xi ).$
ⓘ 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 $m_1\ne m_2$,

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: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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 See also: Annotations for §28.31(iii), §28.31 and Ch.28

More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by

28.31.22		$$\begin{array}{l}{\int }_{u_0}^{u_{\mathrm{\infty }}}{\int }_0^{2\pi }\begin{array}{l}{\mathit{hc}}_{p_1}^{m_1}(u,\xi ){\mathit{hc}}_{p_1}^{m_1}(v,\xi ){\mathit{hc}}_{p_2}^{m_2}(u,\xi )\\ \times {\mathit{hc}}_{p_2}^{m_2}(v,\xi )\left(\cos \left(2u\right)-\cos \left(2v\right)\right)dvdu\end{array}\\ =0,\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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 See also: Annotations for §28.31(iii), §28.31 and Ch.28

and

28.31.23		$$\begin{array}{l}{\int }_{u_0}^{u_{\mathrm{\infty }}}{\int }_0^{2\pi }\begin{array}{l}{\mathit{hs}}_{p_1}^{m_1}(u,\xi ){\mathit{hs}}_{p_1}^{m_1}(v,\xi ){\mathit{hs}}_{p_2}^{m_2}(u,\xi )\\ \times {\mathit{hs}}_{p_2}^{m_2}(v,\xi )\left(\cos \left(2u\right)-\cos \left(2v\right)\right)dvdu\end{array}\\ =0,\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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 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		$$\begin{array}{l}{\int }_{u_0}^{u_{\mathrm{\infty }}}{\int }_0^{2\pi }\begin{array}{l}{\mathit{hc}}_{p_1}^{m_1}(u,\xi ){\mathit{hc}}_{p_1}^{m_1}(v,\xi ){\mathit{hs}}_{p_2}^{m_2}(u,\xi )\\ \times {\mathit{hs}}_{p_2}^{m_2}(v,\xi )\left(\cos \left(2u\right)-\cos \left(2v\right)\right)dvdu\end{array}\\ =0,\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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 See also: Annotations for §28.31(iii), §28.31 and Ch.28

where $(u_0,u_{\mathrm{\infty }})=(0,\mathrm{i}\mathrm{\infty })$ when $\xi >0$, and $(u_0,u_{\mathrm{\infty }})=(\frac{1}{2}\pi ,\frac{1}{2}\pi +\mathrm{i}\mathrm{\infty })$ when .

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