# §29.8 Integral Equations

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

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

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

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

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

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

A special case of (29.8.2) is

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

where

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

Others are:

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

and

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

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