29 Lamé Functions Lamé Functions 29.7 Asymptotic Expansions 29.9 Stability

§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 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 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, : real variable, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.8.E1
Encodings:: TeX, pMML, png

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, : differential of , $\int$ : integral, : real variable, : complex variable, : real parameter, $\nu$ : real parameter, : 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, : 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, : complex variable, : real parameter, : 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 with respect to , : differential of , $\int$ : integral, : nonnegative integer, : real variable, : complex variable, : real parameter, $\nu$ : real parameter and : solution
Permalink:: http://dlmf.nist.gov/29.8.E5
Encodings:: TeX, pMML, png

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, : real variable, : complex variable and : 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 with respect to , : differential of , $\int$ : integral, : nonnegative integer, : real variable, : complex variable, : real parameter, $\nu$ : real parameter and : 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 with respect to , : differential of , $\int$ : integral, : nonnegative integer, : real variable, : complex variable, : real parameter, $\nu$ : real parameter and : 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 with respect to , : differential of , $\int$ : integral, : nonnegative integer, : real variable, : complex variable, : real parameter, $\nu$ : real parameter and : 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 29.7 Asymptotic Expansions 29.9 Stability