§29.8 Integral Equations

Let $w(z)$ be any solution of (29.2.1) of period $4\!K\!$, $w_{2}(z)$ 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 $x=k^{2}\operatorname{sn}\left(z,k\right)\operatorname{sn}\left(z_{1},k\right)% \operatorname{sn}\left(z_{2},k\right)\operatorname{sn}\left(z_{3},k\right)-% \frac{k^{2}}{{k^{\prime}}^{2}}\operatorname{cn}\left(z,k\right)\operatorname{% cn}\left(z_{1},k\right)\operatorname{cn}\left(z_{2},k\right)\operatorname{cn}% \left(z_{3},k\right)+\frac{1}{{k^{\prime}}^{2}}\operatorname{dn}\left(z,k% \right)\operatorname{dn}\left(z_{1},k\right)\operatorname{dn}\left(z_{2},k% \right)\operatorname{dn}\left(z_{3},k\right),$

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 $\mu w(z_{1})w(z_{2})w(z_{3})=\int_{-2\!K\!}^{2\!K\!}\mathsf{P}_{\nu}\left(x% \right)w(z)\mathrm{d}z,$

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 29

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

 29.8.4 $\displaystyle w(z+2\!K\!)$ $\displaystyle=\sigma w(z),$ $\displaystyle w_{2}(z+2\!K\!)$ $\displaystyle=\tau w(z)+\sigma w_{2}(z).$

A special case of (29.8.2) is

 29.8.5 $\mathit{Ec}^{2m}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(\!K\!)-w_{2}(-\!K\!)% }{\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|_{z=0}}=\int_{-\!K\!}^{% \!K\!}\mathsf{P}_{\nu}\left(y\right)\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)% \mathrm{d}z,$

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 29

Others are:

 29.8.7 $\mathit{Ec}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(\!K\!)+w_{2}(-\!K% \!)}{w_{2}(0)}=-k^{2}\operatorname{sn}\left(z_{1},k\right)\int_{-\!K\!}^{\!K\!% }\operatorname{sn}\left(z,k\right)\frac{\mathrm{d}\mathsf{P}_{\nu}\left(y% \right)}{\mathrm{d}y}\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)\mathrm{d}z,$
 29.8.8 $\mathit{Es}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)\frac{\left.\ifrac{\mathrm{d}w% _{2}(z)}{\mathrm{d}z}\right|_{z=\!K\!}+\left.\ifrac{\mathrm{d}w_{2}(z)}{% \mathrm{d}z}\right|_{z=-\!K\!}}{\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}% \right|_{z=0}}=\frac{k^{2}}{k^{\prime}}\operatorname{cn}\left(z_{1},k\right)% \int_{-\!K\!}^{\!K\!}\operatorname{cn}\left(z,k\right)\frac{\mathrm{d}\mathsf{% P}_{\nu}\left(y\right)}{\mathrm{d}y}\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}% \right)\mathrm{d}z,$

and

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

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