§29.8 Integral Equations

Notes:: See Erdélyi et al. (1955, §15.5.3) and Volkmer (1982, 1983).
Keywords:: Ferrers function, Lamé functions, integral equations
Permalink:: http://dlmf.nist.gov/29.8

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

29.8.1 $x=k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)\mathop{\mathrm{sn}\/}\nolimits\left(z_{1},k\right)\mathop{\mathrm{sn}\/}\nolimits\left(z_{2},k\right)\mathop{\mathrm{sn}\/}\nolimits\left(z_{3},k\right)-\frac{k^{2}}{{k^{{\prime}}}^{2}}\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)\mathop{\mathrm{cn}\/}\nolimits\left(z_{1},k\right)\mathop{\mathrm{cn}\/}\nolimits\left(z_{2},k\right)\mathop{\mathrm{cn}\/}\nolimits\left(z_{3},k\right)+\frac{1}{{k^{{\prime}}}^{2}}\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)\mathop{\mathrm{dn}\/}\nolimits\left(z_{1},k\right)\mathop{\mathrm{dn}\/}\nolimits\left(z_{2},k\right)\mathop{\mathrm{dn}\/}\nolimits\left(z_{3},k\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $x$ : real variable, $z$ : complex variable and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.8.E1
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where $z,z_{1},z_{2},z_{3}$ are real, and $\mathop{\mathrm{sn}\/}\nolimits$ , $\mathop{\mathrm{cn}\/}\nolimits$ , $\mathop{\mathrm{dn}\/}\nolimits$ are the Jacobian elliptic functions (§22.2). Then

29.8.2 $\mu w(z_{1})w(z_{2})w(z_{3})=\int _{{-2\!\mathop{K\/}\nolimits\!}}^{{2\!\mathop{K\/}\nolimits\!}}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)w(z)dz,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $x$ : 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
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where $\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)),

29.8.3 $\mu=\frac{2\sigma\tau}{\mathop{\mathscr{W}\/}\nolimits\left\{ w,w_{2}\right\}},$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $w(z)$ : solution, $\mu$ , $\sigma=\pm 1$ and $\tau$
Permalink:: http://dlmf.nist.gov/29.8.E3
Encodings:: TeX, pMML, png

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

29.8.4

$w(z+2\!\mathop{K\/}\nolimits\!)=\sigma w(z),$

$w_{2}(z+2\!\mathop{K\/}\nolimits\!)=\tau w(z)+\sigma w_{2}(z).$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

A special case of (29.8.2) is

29.8.5 $\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z_{1},k^{2}\right)\frac{w_{2}(\!\mathop{K\/}\nolimits\!)-w_{2}(-\!\mathop{K\/}\nolimits\!)}{\left.\ifrac{dw_{2}(z)}{dz}\right|_{{z=0}}}=\int _{{-\!\mathop{K\/}\nolimits\!}}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(y\right)\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)dz,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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, $m$ : 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
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where

29.8.6 $y=\frac{1}{k^{{\prime}}}\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)\mathop{\mathrm{dn}\/}\nolimits\left(z_{1},k\right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : 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

Others are:

29.8.7 $\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z_{1},k^{2}\right)\frac{w_{2}(\!\mathop{K\/}\nolimits\!)+w_{2}(-\!\mathop{K\/}\nolimits\!)}{w_{2}(0)}=-k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(z_{1},k\right)\int _{{-\!\mathop{K\/}\nolimits\!}}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)\frac{d\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(y\right)}{dy}\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)dz,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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, $m$ : 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

29.8.8 $\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z_{1},k^{2}\right)\frac{\left.\ifrac{dw_{2}(z)}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}+\left.\ifrac{dw_{2}(z)}{dz}\right|_{{z=-\!\mathop{K\/}\nolimits\!}}}{\left.\ifrac{dw_{2}(z)}{dz}\right|_{{z=0}}}=\frac{k^{2}}{k^{{\prime}}}\mathop{\mathrm{cn}\/}\nolimits\left(z_{1},k\right)\int _{{-\!\mathop{K\/}\nolimits\!}}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)\frac{d\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(y\right)}{dy}\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)dz,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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, $m$ : 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

and

29.8.9 $\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z_{1},k^{2}\right)\frac{\left.\ifrac{dw_{2}(z)}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}-\left.\ifrac{dw_{2}(z)}{dz}\right|_{{z=-\!\mathop{K\/}\nolimits\!}}}{w_{2}(0)}=-\frac{k^{4}}{k^{{\prime}}}\mathop{\mathrm{sn}\/}\nolimits\left(z_{1},k\right)\mathop{\mathrm{cn}\/}\nolimits\left(z_{1},k\right)\int _{{-\!\mathop{K\/}\nolimits\!}}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)\frac{{d}^{2}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(y\right)}{{dy}^{2}}\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)dz.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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, $m$ : 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

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