29 Lamé Functions Lamé Functions 29.7 Asymptotic Expansions 29.9 Stability

§29.8 Integral Equations

ⓘ

Keywords:: Ferrers function, Lamé functions, integral equation for Lamé functions, integral equations, of the first kind
Notes:: See Erdélyi et al. (1955, §15.5.3) and Volkmer (1982, 1983).
Permalink:: http://dlmf.nist.gov/29.8
See also:: Annotations for Ch.29

Let $w(z)$ be any solution of (29.2.1) of period $4K$ , $w_{2}(z)$ be a linearly independent solution, and $\mathscr{W}\left\{w,w_{2}\right\}$ denote their Wronskian. Also let $x$ be defined by

29.8.1		$x=k^{2}\operatorname{sn}\left(z,k\right)\operatorname{sn}\left(z_{1},k\right)% \operatorname{sn}\left(z_{2},k\right)\operatorname{sn}\left(z_{3},k\right)-% \frac{k^{2}}{{k^{\prime}}^{2}}\operatorname{cn}\left(z,k\right)\operatorname{% cn}\left(z_{1},k\right)\operatorname{cn}\left(z_{2},k\right)\operatorname{cn}% \left(z_{3},k\right)+\frac{1}{{k^{\prime}}^{2}}\operatorname{dn}\left(z,k% \right)\operatorname{dn}\left(z_{1},k\right)\operatorname{dn}\left(z_{2},k% \right)\operatorname{dn}\left(z_{3},k\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $x$ : real variable, $z$ : complex variable and $k$ : real parameter Permalink: http://dlmf.nist.gov/29.8.E1 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

where $z,z_{1},z_{2},z_{3}$ are real, and $\operatorname{sn}$ , $\operatorname{cn}$ , $\operatorname{dn}$ are the Jacobian elliptic functions (§22.2). Then

29.8.2		$\mu w(z_{1})w(z_{2})w(z_{3})=\int_{-2K}^{2K}\mathsf{P}_{\nu}\left(x\right)w(z)% \,\mathrm{d}z,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter, $w(z)$ : solution and $\mu$ Referenced by: §29.8 Permalink: http://dlmf.nist.gov/29.8.E2 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

where $\mathsf{P}_{\nu}\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)),

29.8.3		$\mu=\frac{2\sigma\tau}{\mathscr{W}\left\{w,w_{2}\right\}},$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $w(z)$ : solution, $\mu$ , $\sigma=\pm 1$ and $\tau$ Permalink: http://dlmf.nist.gov/29.8.E3 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

and $\sigma$ (= $\pm 1$ ) and $\tau$ are determined by

29.8.4	$\displaystyle w(z+2K)$	$\displaystyle=\sigma w(z),$
29.8.4	$\displaystyle w_{2}(z+2K)$	$\displaystyle=\tau w(z)+\sigma w_{2}(z).$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex variable, $k$ : real parameter, $w(z)$ : solution, $\sigma=\pm 1$ and $\tau$ Permalink: http://dlmf.nist.gov/29.8.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.8 and Ch.29

A special case of (29.8.2) is

29.8.5		$\mathit{Ec}^{2m}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(K)-w_{2}(-K)}{\left.% \ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right\|_{z=0}}=\int_{-K}^{K}\mathsf{P}_% {\nu}\left(y\right)\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)\,\mathrm{d}z,$
ⓘ Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $m$ : nonnegative integer, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution Permalink: http://dlmf.nist.gov/29.8.E5 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

where

29.8.6		$y=\frac{1}{k^{\prime}}\operatorname{dn}\left(z,k\right)\operatorname{dn}\left(% z_{1},k\right).$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $y$ : real variable, $z$ : complex variable and $k$ : real parameter Permalink: http://dlmf.nist.gov/29.8.E6 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

Others are:

29.8.7		$\mathit{Ec}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(K)+w_{2}(-K)}{w_{2% }(0)}=-k^{2}\operatorname{sn}\left(z_{1},k\right)\int_{-K}^{K}\operatorname{sn% }\left(z,k\right)\frac{\mathrm{d}\mathsf{P}_{\nu}\left(y\right)}{\mathrm{d}y}% \mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)\,\mathrm{d}z,$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $m$ : nonnegative integer, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution Permalink: http://dlmf.nist.gov/29.8.E7 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

29.8.8		$\mathit{Es}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)\frac{\left.\ifrac{\mathrm{d}w% _{2}(z)}{\mathrm{d}z}\right\|_{z=K}+\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}% z}\right\|_{z=-K}}{\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right\|_{z=0}}=% \frac{k^{2}}{k^{\prime}}\operatorname{cn}\left(z_{1},k\right)\int_{-K}^{K}% \operatorname{cn}\left(z,k\right)\frac{\mathrm{d}\mathsf{P}_{\nu}\left(y\right% )}{\mathrm{d}y}\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)\,\mathrm{d}z,$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $m$ : nonnegative integer, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution Permalink: http://dlmf.nist.gov/29.8.E8 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

and

29.8.9		$\mathit{Es}^{2m+2}_{\nu}\left(z_{1},k^{2}\right)\frac{\left.\ifrac{\mathrm{d}w% _{2}(z)}{\mathrm{d}z}\right\|_{z=K}-\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}% z}\right\|_{z=-K}}{w_{2}(0)}=-\frac{k^{4}}{k^{\prime}}\operatorname{sn}\left(z_% {1},k\right)\operatorname{cn}\left(z_{1},k\right)\int_{-K}^{K}\operatorname{sn% }\left(z,k\right)\operatorname{cn}\left(z,k\right)\frac{{\mathrm{d}}^{2}% \mathsf{P}_{\nu}\left(y\right)}{{\mathrm{d}y}^{2}}\mathit{Es}^{2m+2}_{\nu}% \left(z,k^{2}\right)\,\mathrm{d}z.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $m$ : nonnegative integer, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution Permalink: http://dlmf.nist.gov/29.8.E9 Encodings: TeX, pMML, png See also: Annotations for §29.8 and Ch.29

For further integral equations see Arscott (1964a), Erdélyi et al. (1955, §15.5.3), Shail (1980), Sleeman (1968a), and Volkmer (1982, 1983, 1984).

29.7 Asymptotic Expansions 29.9 Stability

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§29.8 Integral Equations

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