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⇑ 29 Lamé Functions
⇑ Lamé Functions
⇐ 29.5 Special Cases and Limiting Forms29.7 Asymptotic Expansions ⇒

§29.6 Fourier Series

Contents

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

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

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.

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, \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).

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,
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Symbols:
\alpha _{p}, \beta _{p}, H 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,
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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

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},

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,
29.6.13\tfrac{1}{2}C_{0}+\sum _{{p=1}}^{{\infty}}C_{{2p}}>0,
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,
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.

§29.6(ii) Function \mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\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

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),

where

29.6.24(\beta _{0}-H)C_{1}+\alpha _{0}C_{3}=0,
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Symbols:
\alpha _{p}, \beta _{p}, H 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,
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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

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},

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},
29.6.28\sum _{{p=0}}^{{\infty}}C_{{2p+1}}>0,
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,
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.

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

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

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),

where

29.6.39(\beta _{0}-H)D_{1}+\alpha _{0}D_{3}=0,
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Symbols:
\alpha _{p}, \beta _{p}, D_{{2p+1}}: coefficents and H
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,
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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

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},

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},
29.6.43\sum _{{p=0}}^{{\infty}}(2p+1)D_{{2p+1}}>0,
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,
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.

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

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

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),

where

29.6.54(\beta _{0}-H)D_{2}+\alpha _{0}D_{4}=0,
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Symbols:
\alpha _{p}, \beta _{p}, H 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,
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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

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},

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,
29.6.58\sum _{{p=0}}^{{\infty}}(2p+2)D_{{2p+2}}>0,
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,
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.
© 2010–2012 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.5; Release date 2012-10-01 Math-Image version (See MathML) . A Printed Companion is available. ⇐ 29.5 Special Cases and Limiting Forms29.7 Asymptotic Expansions ⇒