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

§29.6 Fourier Series

ⓘ

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

ⓘ

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		$\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{% \infty}A_{2p}\cos\left(2p\phi\right).$
ⓘ Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cos\NVar{z}$ : cosine function, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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		$H=2a^{2m}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2},$
ⓘ Defines: $H$ (locally) Symbols: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : 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		$(\beta_{0}-H)A_{0}+\alpha_{0}A_{2}=0,$
ⓘ Symbols: $\alpha_{p}$ , $\beta_{p}$ , $H$ 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		$\gamma_{p}A_{2p-2}+(\beta_{p}-H)A_{2p}+\alpha_{p}A_{2p+2}=0,$
$p\geq 1$ ,
ⓘ Symbols: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ 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		$\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2}=1,$
ⓘ Symbols: $p$ : 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		$\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p}>0.$
ⓘ Symbols: $p$ : 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 $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ 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 $m>n$ .
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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 $H$ 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		$\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}\cos\left(2p\phi\right)\right).$
ⓘ 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, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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		$(\beta_{0}-H)C_{0}+\alpha_{0}C_{2}=0,$
ⓘ Symbols: $\alpha_{p}$ , $\beta_{p}$ , $H$ 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		$\gamma_{p}C_{2p-2}+(\beta_{p}-H)C_{2p}+\alpha_{p}C_{2p+2}=0,$
$p\geq 1$ ,
ⓘ Symbols: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ 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	$\displaystyle\alpha_{p}$	$\displaystyle=\begin{cases}\nu(\nu+1)k^{2},&p=0,\\ \frac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},&p\geq 1,\end{cases}$
	$\displaystyle\beta_{p}$	$\displaystyle=4p^{2}(2-k^{2}),$
	$\displaystyle\gamma_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2},$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : 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		$\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty% }C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2}=1,$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : 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		$\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}>0,$
ⓘ Symbols: $p$ : 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 $m>n$ ,
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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 $x$ , $\int$ : integral, $m$ : nonnegative integer, $p$ : nonnegative integer, $x$ : real variable, $k$ : 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 $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$

ⓘ

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		$\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)=\sum_{p=0}^{\infty}A_{2p+1}\cos% \left((2p+1)\phi\right).$
ⓘ Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cos\NVar{z}$ : cosine function, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $H$ (locally) Symbols: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : 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: $\alpha_{p}$ , $\beta_{p}$ , $H$ 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: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ 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		$\sum_{p=0}^{\infty}A_{2p+1}^{2}=1,$
ⓘ Symbols: $p$ : 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		$\sum_{p=0}^{\infty}A_{2p+1}>0,$
ⓘ Symbols: $p$ : 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 $m>n$ .
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $\alpha_{p}$ , $\beta_{p}$ , $H$ 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: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ 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	$\displaystyle\alpha_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$
	$\displaystyle\beta_{p}$	$\displaystyle=\begin{cases}2-k^{2}+\frac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$
	$\displaystyle\gamma_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : 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: $p$ : nonnegative integer, $k$ : 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: $p$ : 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 $m>n$ ,
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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 $x$ , $\int$ : integral, $m$ : nonnegative integer, $p$ : nonnegative integer, $x$ : real variable, $k$ : 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, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $H$ (locally) Symbols: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : 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: $\alpha_{p}$ , $\beta_{p}$ , $H$ 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: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ 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: $p$ : 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: $p$ : 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 $m>n$ .
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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 $H$ 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: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $D_{2p+1}$ : coefficents and $H$ 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	$\displaystyle\alpha_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$
	$\displaystyle\beta_{p}$	$\displaystyle=\begin{cases}2-k^{2}-\frac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$
	$\displaystyle\gamma_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : 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		$\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}D_{2p+1}^{2}+{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{\infty}D_{2p+1}D_{2p+3}\right)=1},$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : 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		$\sum_{p=0}^{\infty}(2p+1)D_{2p+1}>0,$
ⓘ Symbols: $p$ : 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 $m>n$ ,
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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		$\sum_{p=0}^{\infty}B_{2p+1}D_{2p+1}=\frac{4}{\pi}\int_{0}^{K}\left(\mathit{Es}% ^{2m+1}_{\nu}\left(x,k^{2}\right)\right)^{2}\,\mathrm{d}x.$
ⓘ 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 $x$ , $\int$ : integral, $m$ : nonnegative integer, $p$ : nonnegative integer, $x$ : real variable, $k$ : 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 $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$

ⓘ

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

29.6.46		$\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=\sum_{p=1}^{\infty}B_{2p}\sin% \left(2p\phi\right).$
ⓘ Symbols: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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.E46 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

Here

29.6.47		$H=2b^{2m+2}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2},$
ⓘ Defines: $H$ (locally) Symbols: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter Permalink: http://dlmf.nist.gov/29.6.E47 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.48		$(\beta_{0}-H)B_{2}+\alpha_{0}B_{4}=0,$
ⓘ Symbols: $\alpha_{p}$ , $\beta_{p}$ , $H$ and $B_{2p+1}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E48 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.49		$\gamma_{p}B_{2p}+(\beta_{p}-H)B_{2p+2}+\alpha_{p}B_{2p+4}=0,$
$p\geq 1$ ,
ⓘ Symbols: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ and $B_{2p+1}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E49 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

with $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.17), and

29.6.50		$\sum_{p=1}^{\infty}B_{2p}^{2}=1,$
ⓘ Symbols: $p$ : nonnegative integer and $B_{2p+1}$ : coefficents Referenced by: §29.6(iv) Permalink: http://dlmf.nist.gov/29.6.E50 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.51		$\sum_{p=0}^{\infty}(2p+2)B_{2p+2}>0,$
ⓘ Symbols: $p$ : nonnegative integer and $B_{2p+1}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E51 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.52		$\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$
$\nu\neq 2n+2$ , or $\nu=2n+2$ and $m>n$ .
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $B_{2p+1}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E52 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

Also,

29.6.53		$\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=1}^{\infty}D_{2p}\sin\left(2p\phi\right),$
ⓘ 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, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $D_{2p}$ : coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E53 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

where

29.6.54		$(\beta_{0}-H)D_{2}+\alpha_{0}D_{4}=0,$
ⓘ Symbols: $\alpha_{p}$ , $\beta_{p}$ , $H$ and $D_{2p}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E54 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.55		$\gamma_{p}D_{2p}+(\beta_{p}-H)D_{2p+2}+\alpha_{p}D_{2p+4}=0,$
$p\geq 1$ ,
ⓘ Symbols: $p$ : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , $H$ and $D_{2p}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E55 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

with

29.6.56	$\displaystyle\alpha_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$
	$\displaystyle\beta_{p}$	$\displaystyle=(2p+2)^{2}(2-k^{2}),$
	$\displaystyle\gamma_{p}$	$\displaystyle=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter, $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$ Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E56 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

and

29.6.57		$\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2% }\sum_{p=1}^{\infty}D_{2p}D_{2p+2}=1,$
ⓘ Symbols: $p$ : nonnegative integer, $k$ : real parameter and $D_{2p}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E57 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.58		$\sum_{p=0}^{\infty}(2p+2)D_{2p+2}>0,$
ⓘ Symbols: $p$ : nonnegative integer and $D_{2p}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E58 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.59		$\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$
$\nu\neq 2n+3$ , or $\nu=2n+3$ and $m>n$ ,
ⓘ Symbols: $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $D_{2p}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E59 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.60		$\sum_{p=1}^{\infty}B_{2p}D_{2p}=\frac{4}{\pi}\int_{0}^{K}\left(\mathit{Es}^{2m% +2}_{\nu}\left(x,k^{2}\right)\right)^{2}\,\mathrm{d}x.$
ⓘ 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 $x$ , $\int$ : integral, $m$ : nonnegative integer, $p$ : nonnegative integer, $x$ : real variable, $k$ : real parameter, $\nu$ : real parameter, $B_{2p+1}$ : coefficents and $D_{2p}$ : coefficents Permalink: http://dlmf.nist.gov/29.6.E60 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.5 Special Cases and Limiting Forms 29.7 Asymptotic Expansions

NIST

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§29.6 Fourier Series

Contents

§29.6(i) Function 𝐸𝑐ν2⁢m⁡(z,k2)

§29.6(ii) Function 𝐸𝑐ν2⁢m+1⁡(z,k2)

§29.6(iii) Function 𝐸𝑠ν2⁢m+1⁡(z,k2)

§29.6(iv) Function 𝐸𝑠ν2⁢m+2⁡(z,k2)

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