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), §29.21, §29.3(vi)
Permalink:: http://dlmf.nist.gov/29.6

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

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

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

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

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

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\cos\/}\nolimits z$ : 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

Here

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and
Referenced by:: §29.6(i)
Permalink:: http://dlmf.nist.gov/29.6.E2
Encodings:: TeX, pMML, png

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

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

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

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

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:: : 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

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

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

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

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

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\cos\/}\nolimits z$ : 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

Here $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ is as in §22.2, and

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

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $C_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E9
Encodings:: TeX, pMML, png

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

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

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

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:: : nonnegative integer, : real parameter and $C_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E12
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer and $C_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E13
Encodings:: TeX, pMML, png

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

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

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

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : 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

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

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

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

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\cos\/}\nolimits z$ : 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

Here

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and
Permalink:: http://dlmf.nist.gov/29.6.E17
Encodings:: TeX, pMML, png

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

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

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

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

Symbols:: : nonnegative integer and $A_{{2p}}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.6.E21
Encodings:: TeX, pMML, png

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

Also,

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\cos\/}\nolimits z$ : 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

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

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

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

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

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

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

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

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : 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

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

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

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

Symbols:: $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\sin\/}\nolimits z$ : 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

Here

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

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and
Permalink:: http://dlmf.nist.gov/29.6.E32
Encodings:: TeX, pMML, png

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

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

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

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

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

Also,

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\sin\/}\nolimits z$ : 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

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

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

with

29.6.41

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

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

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

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

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:: : nonnegative integer, : real parameter and $D_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E42
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer and $D_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E43
Encodings:: TeX, pMML, png

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

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

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

Symbols:: $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : 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

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

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

29.6.46 $\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)=\sum_{{% p=1}}^{{\infty}}B_{{2p}}\mathop{\sin\/}\nolimits\!\left(2p\phi\right).$

Symbols:: $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\sin\/}\nolimits z$ : 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.E46
Encodings:: TeX, pMML, png

Here

29.6.47 $H=2\!\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2},$

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and
Permalink:: http://dlmf.nist.gov/29.6.E47
Encodings:: TeX, pMML, png

29.6.48 $(\beta_{0}-H)B_{2}+\alpha_{0}B_{4}=0,$

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $B_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E48
Encodings:: TeX, pMML, png

29.6.49 $\gamma_{p}B_{{2p}}+(\beta_{p}-H)B_{{2p+2}}+\alpha_{p}B_{{2p+4}}=0,$ $p\geq 1$ ,

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $B_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E49
Encodings:: TeX, pMML, png

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:: : nonnegative integer and $B_{{2p+1}}$ : coefficents
Referenced by:: §29.6(iv)
Permalink:: http://dlmf.nist.gov/29.6.E50
Encodings:: TeX, pMML, png

29.6.51 $\sum_{{p=0}}^{{\infty}}(2p+2)B_{{2p+2}}>0,$

Symbols:: : nonnegative integer and $B_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E51
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $B_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E52
Encodings:: TeX, pMML, png

Also,

29.6.53 $\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)=\mathop% {\mathrm{dn}\/}\nolimits\left(z,k\right)\sum_{{p=1}}^{{\infty}}D_{{2p}}\mathop% {\sin\/}\nolimits\!\left(2p\phi\right),$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer, : complex variable, : 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

where

29.6.54 $(\beta_{0}-H)D_{2}+\alpha_{0}D_{4}=0,$

Symbols:: $\alpha_{p}$ , $\beta_{p}$ , and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E54
Encodings:: TeX, pMML, png

29.6.55 $\gamma_{p}D_{{2p}}+(\beta_{p}-H)D_{{2p+2}}+\alpha_{p}D_{{2p+4}}=0,$ $p\geq 1$ ,

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ , and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E55
Encodings:: TeX, pMML, png

with

29.6.56

$\alpha_{p}=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$

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

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

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.E56
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

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:: : nonnegative integer, : real parameter and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E57
Encodings:: TeX, pMML, png

29.6.58 $\sum_{{p=0}}^{{\infty}}(2p+2)D_{{2p+2}}>0,$

Symbols:: : nonnegative integer and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E58
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer, : nonnegative integer, : nonnegative integer, : real parameter, $\nu$ : real parameter and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E59
Encodings:: TeX, pMML, png

29.6.60 $\sum_{{p=1}}^{{\infty}}B_{{2p}}D_{{2p}}=\frac{4}{\pi}\int_{0}^{{\!\mathop{K\/}% \nolimits\!}}\left(\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(x,k% ^{2}\right)\right)^{2}dx.$

Symbols:: $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : real variable, : 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

© 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. 29.5 Special Cases and Limiting Forms 29.7 Asymptotic Expansions

§29.6 Fourier Series

Contents

§29.6(i) Function

§29.6(ii) Function

§29.6(iii) Function

§29.6(iv) Function

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

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

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

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