§29.8 Integral Equations

Let $w(z)$ be any solution of (29.2.1) of period $4K$, $w_2(z)$ be a linearly independent solution, and $𝒲\left\{w,w_2\right\}$ denote their Wronskian. Also let $x$ be defined by

29.8.1		$$x=\begin{array}{l}\begin{array}{l}{k}^2\mathrm{sn}(z,k)\mathrm{sn}(z_1,k)\mathrm{sn}(z_2,k)\mathrm{sn}(z_3,k)\\ -\frac{k^2}{k^{\prime }{}^2}\mathrm{cn}(z,k)\mathrm{cn}(z_1,k)\mathrm{cn}(z_2,k)\mathrm{cn}(z_3,k)\end{array}\\ +\frac{1}{k^{\prime }{}^2}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k)\mathrm{dn}(z_2,k)\mathrm{dn}(z_3,k),\end{array}$$
ⓘ Symbols: $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ are the Jacobian elliptic functions (§22.2). Then

29.8.2		$$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{\mathsf{P}}_{\nu }\left(x\right)w(z)dz,$$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		$$\mu =\frac{2\sigma \tau }{𝒲\left\{w,w_2\right\}},$$
ⓘ Symbols: $𝒲$: 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		$w(z+2K)$	$=\sigma w(z),$
		$w_2(z+2K)$	$=\tau w(z)+\sigma w_2(z).$
ⓘ Symbols: $K\left(k\right)$: 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		$${\mathit{Ec}}_{\nu }^{2m}(z_1,k^2)\frac{w_2(K)-w_2(-K)}{{d{w}_2(z)/dz|}_{z=0}}={\int }_{-K}^K{\mathsf{P}}_{\nu }\left(y\right){\mathit{Ec}}_{\nu }^{2m}(z,k^2)dz,$$
ⓘ Symbols: ${\mathit{Ec}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		$$y=\frac{1}{k^{\prime }}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k).$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: 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		$$\begin{array}{l}{\mathit{Ec}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}\\ =-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{\mathsf{P}}_{\nu }\left(y\right)}{dy}{\mathit{Ec}}_{\nu }^{2m+1}(z,k^2)dz,\end{array}$$
ⓘ Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathit{Ec}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		$$\begin{array}{l}{\mathit{Es}}_{\nu }^{2m+1}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}+{d{w}_2(z)/dz|}_{z=-K}}{{d{w}_2(z)/dz|}_{z=0}}\\ =\frac{k^2}{k^{\prime }}\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{cn}(z,k)\frac{d{\mathsf{P}}_{\nu }\left(y\right)}{dy}{\mathit{Es}}_{\nu }^{2m+1}(z,k^2)dz,\end{array}$$
ⓘ Symbols: $\mathrm{cn}(z,k)$: Jacobian elliptic function, ${\mathit{Es}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		$$\begin{array}{l}{\mathit{Es}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}\\ =-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{\mathsf{P}}_{\nu }\left(y\right)}{{dy}^2}{\mathit{Es}}_{\nu }^{2m+2}(z,k^2)dz.\end{array}$$
ⓘ Symbols: $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathit{Es}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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).