29 Lamé Functions Lamé Functions 29.5 Special Cases and Limiting Forms 29.7 Asymptotic Expansions

§29.6 Fourier Series

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Keywords:: Fourier series, Lamé functions
Referenced by:: §29.15(i), 2nd item, §29.3(vi)
Permalink:: http://dlmf.nist.gov/29.6
See also:: Annotations for Ch.29

§29.6(i) Function $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$
§29.6(ii) Function $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$
§29.6(iii) Function $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$
§29.6(iv) Function $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$

§29.6(i) Function $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$

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Keywords:: Lamé functions, Lamé polynomials, relation to Lamé functions, relations to Lamé polynomials
Notes:: See Erdélyi et al. (1955, §15.5.1) and Ince (1940b). The normalization formula (29.6.5) is adopted from Jansen (1977).
Permalink:: http://dlmf.nist.gov/29.6.i
See also:: Annotations for §29.6 and Ch.29

With $\phi=\frac{1}{2}\pi-\operatorname{am}\left(z,k\right)$ , as in (29.2.5), we have

29.6.1 $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{% \infty}A_{2p}\cos\left(2p\phi\right).$

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Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

Here

29.6.2 $H=2a^{2m}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2},$

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Defines:: (locally)
Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Referenced by:: §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.3 $(\beta_{0}-H)A_{0}+\alpha_{0}A_{2}=0,$

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Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.20(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.4 $\gamma_{p}A_{2p-2}+(\beta_{p}-H)A_{2p}+\alpha_{p}A_{2p+2}=0,$ $p\geq 1$ ,

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Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.20(i), §29.6(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

with $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.11) and (29.3.12), and

29.6.5 $\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2}=1,$

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Symbols:: : nonnegative integer and $A_{2p}$ : coefficients
Referenced by:: §29.20(i), §29.6(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.6 $\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p}>0.$

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Symbols:: : nonnegative integer and $A_{2p}$ : coefficients
Referenced by:: §29.20(i), §29.6(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

When $\nu\neq 2n$ , where is a nonnegative integer, it follows from §2.9(i) that for any value of the system (29.6.4)–(29.6.6) has a unique recessive solution $A_{0},A_{2},A_{4},\dots$ ; furthermore

29.6.7 $\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n$ , or $\nu=2n$ and

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Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.6.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

In addition, if satisfies (29.6.2), then (29.6.3) applies.

In the special case $\nu=2n$ , $m=0,1,\dots,n$ , there is a unique nontrivial solution with the property $A_{2p}=0$ , $p=n+1,n+2,\dots$ . This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$ , followed by normalization via (29.6.5) and (29.6.6). Consequently, $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i).

An alternative version of the Fourier series expansion (29.6.1) is given by

29.6.8 $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}\cos\left(2p\phi\right)\right).$

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $C_{2p}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

Here $\operatorname{dn}\left(z,k\right)$ is as in §22.2, and

29.6.9 $(\beta_{0}-H)C_{0}+\alpha_{0}C_{2}=0,$

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Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.10 $\gamma_{p}C_{2p-2}+(\beta_{p}-H)C_{2p}+\alpha_{p}C_{2p+2}=0,$ $p\geq 1$ ,

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Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

with $\alpha_{p},\beta_{p}$ , and $\gamma_{p}$ now defined by

29.6.11

$\alpha_{p}=\begin{cases}\nu(\nu+1)k^{2},&p=0,\\ \frac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},&p\geq 1,\end{cases}$

$\beta_{p}=4p^{2}(2-k^{2}),$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2},$

ⓘ

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E11
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

and

29.6.12 $\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty% }C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2}=1,$

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Symbols:: : nonnegative integer, : real parameter and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.13 $\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}>0,$

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Symbols:: : nonnegative integer and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.14 $\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+1$ , or $\nu=2n+1$ and

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Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

29.6.15 $\tfrac{1}{2}A_{0}C_{0}+\sum_{p=1}^{\infty}A_{2p}C_{2p}=\frac{4}{\pi}\int_{0}^{% K}\left(\mathit{Ec}^{2m}_{\nu}\left(x,k^{2}\right)\right)^{2}\,\mathrm{d}x.$

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : real variable, : real parameter, $\nu$ : real parameter, $A_{2p}$ : coefficients and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(i), §29.6 and Ch.29

§29.6(ii) Function $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$

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Notes:: See Erdélyi et al. (1955, §15.5.1) and Ince (1940b). The normalization formula (29.6.20) is adopted from Jansen (1977).
Permalink:: http://dlmf.nist.gov/29.6.ii
See also:: Annotations for §29.6 and Ch.29

29.6.16 $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)=\sum_{p=0}^{\infty}A_{2p+1}\cos% \left((2p+1)\phi\right).$

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Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $A_{2p}$ : coefficients
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

Here

29.6.17 $H=2a^{2m+1}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2},$

ⓘ

Defines:: (locally)
Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.6.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.18 $(\beta_{0}-H)A_{1}+\alpha_{0}A_{3}=0,$

ⓘ

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.6.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.19 $\gamma_{p}A_{2p-1}+(\beta_{p}-H)A_{2p+1}+\alpha_{p}A_{2p+3}=0,$ $p\geq 1$ ,

ⓘ

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.6.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

with $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.13) and (29.3.14), and

29.6.20 $\sum_{p=0}^{\infty}A_{2p+1}^{2}=1,$

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Symbols:: : nonnegative integer and $A_{2p}$ : coefficients
Referenced by:: §29.6(ii)
Permalink:: http://dlmf.nist.gov/29.6.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.21 $\sum_{p=0}^{\infty}A_{2p+1}>0,$

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Symbols:: : nonnegative integer and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.6.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.22 $\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+1$ , or $\nu=2n+1$ and

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.6.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

Also,

29.6.23 $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=0}^{\infty}C_{2p+1}\cos\left((2p+1)\phi\right),$

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $C_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

where

29.6.24 $(\beta_{0}-H)C_{1}+\alpha_{0}C_{3}=0,$

ⓘ

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.25 $\gamma_{p}C_{2p-1}+(\beta_{p}-H)C_{2p+1}+\alpha_{p}C_{2p+3}=0,$ $p\geq 1$ ,

ⓘ

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

with

29.6.26

$\alpha_{p}=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$

$\beta_{p}=\begin{cases}2-k^{2}+\frac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},$

ⓘ

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E26
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

and

29.6.27 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}C_{2p+1}^{2}-{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}C_{1}^{2}+\sum_{p=0}^{\infty}C_{2p+1}C_{2p+3}\right)=1},$

ⓘ

Symbols:: : nonnegative integer, : real parameter and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.28 $\sum_{p=0}^{\infty}C_{2p+1}>0,$

ⓘ

Symbols:: : nonnegative integer and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.29 $\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+2$ , or $\nu=2n+2$ and

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.30 $\sum_{p=0}^{\infty}A_{2p+1}C_{2p+1}=\frac{4}{\pi}\int_{0}^{K}\left(\mathit{Ec}% ^{2m+1}_{\nu}\left(x,k^{2}\right)\right)^{2}\,\mathrm{d}x.$

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : real variable, : real parameter, $\nu$ : real parameter, $A_{2p}$ : coefficients and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E30
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(ii), §29.6 and Ch.29

§29.6(iii) Function $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$

ⓘ

Notes:: See Erdélyi et al. (1955, §15.5.1) and Ince (1940b). The normalization formula (29.6.35) is adopted from Jansen (1977).
Permalink:: http://dlmf.nist.gov/29.6.iii
See also:: Annotations for §29.6 and Ch.29

29.6.31 $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)=\sum_{p=0}^{\infty}B_{2p+1}\sin% \left((2p+1)\phi\right).$

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\sin\NVar{z}$ : sine function, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $B_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

Here

29.6.32 $H=2b^{2m+1}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2},$

ⓘ

Defines:: (locally)
Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.6.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.33 $(\beta_{0}-H)B_{1}+\alpha_{0}B_{3}=0,$

ⓘ

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.34 $\gamma_{p}B_{2p-1}+(\beta_{p}-H)B_{2p+1}+\alpha_{p}B_{2p+3}=0,$ $p\geq 1$ ,

ⓘ

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

with $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.15), (29.3.16), and

29.6.35 $\sum_{p=0}^{\infty}B_{2p+1}^{2}=1,$

ⓘ

Symbols:: : nonnegative integer and $B_{2p+1}$ : coefficents
Referenced by:: §29.6(iii)
Permalink:: http://dlmf.nist.gov/29.6.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.36 $\sum_{p=0}^{\infty}(2p+1)B_{2p+1}>0,$

ⓘ

Symbols:: : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.37 $\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+1$ , or $\nu=2n+1$ and

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

Also,

29.6.38 $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=0}^{\infty}D_{2p+1}\sin\left((2p+1)\phi\right),$

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\sin\NVar{z}$ : sine function, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $D_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

where

29.6.39 $(\beta_{0}-H)D_{1}+\alpha_{0}D_{3}=0,$

ⓘ

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , $D_{2p+1}$ : coefficents and
Permalink:: http://dlmf.nist.gov/29.6.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.40 $\gamma_{p}D_{2p-1}+(\beta_{p}-H)D_{2p+1}+\alpha_{p}D_{2p+3}=0,$ $p\geq 1$ ,

ⓘ

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $D_{2p+1}$ : coefficents and
Permalink:: http://dlmf.nist.gov/29.6.E40
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

with

29.6.41

$\alpha_{p}=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$

$\beta_{p}=\begin{cases}2-k^{2}-\frac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},$

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Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E41
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

and

29.6.42 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}D_{2p+1}^{2}+{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{\infty}D_{2p+1}D_{2p+3}\right)=1},$

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Symbols:: : nonnegative integer, : real parameter and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E42
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.43 $\sum_{p=0}^{\infty}(2p+1)D_{2p+1}>0,$

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Symbols:: : nonnegative integer and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E43
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.44 $\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+2$ , or $\nu=2n+2$ and

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Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E44
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.45 $\sum_{p=0}^{\infty}B_{2p+1}D_{2p+1}=\frac{4}{\pi}\int_{0}^{K}\left(\mathit{Es}% ^{2m+1}_{\nu}\left(x,k^{2}\right)\right)^{2}\,\mathrm{d}x.$

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Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : real variable, : real parameter, $\nu$ : real parameter, $D_{2p+1}$ : coefficents and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E45
Encodings:: TeX, pMML, png
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

§29.6(iv) Function $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$

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Keywords:: Fourier series, Lamé functions
Notes:: See Erdélyi et al. (1955, §15.5.1) and Ince (1940b). The normalization formula (29.6.50) is adopted from Jansen (1977).
Permalink:: http://dlmf.nist.gov/29.6.iv
See also:: Annotations for §29.6 and Ch.29