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26 <div class=
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27 <a name=
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"link" href=
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"Overvew of the Jacobi Elliptic Functions">Overvew of the Jacobi Elliptic
29 </h3></div></div></div>
31 There are twelve Jacobi Elliptic functions, of which the three copolar functions
32 <span class=
"emphasis"><em>sn
</em></span>,
<span class=
"emphasis"><em>cn
</em></span> and
<span class=
"emphasis"><em>dn
</em></span>
33 are the most important as the other nine can be computed from these three
34 <a href=
"#ftn.math_toolkit.jacobi.jac_over.f0" class=
"footnote"><sup class=
"footnote"><a name=
"math_toolkit.jacobi.jac_over.f0"></a>[
2]
</sup></a> <a href=
"#ftn.math_toolkit.jacobi.jac_over.f1" class=
"footnote"><sup class=
"footnote"><a name=
"math_toolkit.jacobi.jac_over.f1"></a>[
3]
</sup></a> <a href=
"#ftn.math_toolkit.jacobi.jac_over.f2" class=
"footnote"><sup class=
"footnote"><a name=
"math_toolkit.jacobi.jac_over.f2"></a>[
4]
</sup></a>.
37 These functions each take two arguments: a parameter, and a variable as described
41 Like all elliptic functions these can be parameterised in a number of ways:
43 <div class=
"itemizedlist"><ul class=
"itemizedlist" style=
"list-style-type: disc; ">
45 In terms of a parameter
<span class=
"emphasis"><em>m
</em></span>.
48 In terms of the elliptic modulus
<span class=
"emphasis"><em>k
</em></span> where
<span class=
"emphasis"><em>m
49 = k
<sup>2</sup></em></span>.
52 In terms of the modular angle
α, where
<span class=
"emphasis"><em>m = sin
<sup>2</sup>α</em></span>.
56 In our implementation, these functions all take the elliptic modulus
<span class=
"emphasis"><em>k
</em></span>
60 In addition the variable
<span class=
"emphasis"><em>u
</em></span> is sometimes expressed as
61 an amplitude
φ, in our implementation we always use
<span class=
"emphasis"><em>u
</em></span>.
64 Finally note that our functions all take the elliptic modulus as the first
65 argument - this is for alignment with the Elliptic Integrals.
68 There are twenve functions for computing the twelve individual Jacobi elliptic
69 functions:
<a class=
"link" href=
"jacobi_cd.html" title=
"Jacobi Elliptic Function cd">jacobi_cd
</a>,
70 <a class=
"link" href=
"jacobi_cn.html" title=
"Jacobi Elliptic Function cn">jacobi_cn
</a>,
<a class=
"link" href=
"jacobi_cs.html" title=
"Jacobi Elliptic Function cs">jacobi_cs
</a>,
71 <a class=
"link" href=
"jacobi_dc.html" title=
"Jacobi Elliptic Function dc">jacobi_dc
</a>,
<a class=
"link" href=
"jacobi_dn.html" title=
"Jacobi Elliptic Function dn">jacobi_dn
</a>,
72 <a class=
"link" href=
"jacobi_ds.html" title=
"Jacobi Elliptic Function ds">jacobi_ds
</a>,
<a class=
"link" href=
"jacobi_nc.html" title=
"Jacobi Elliptic Function nc">jacobi_nc
</a>,
73 <a class=
"link" href=
"jacobi_nd.html" title=
"Jacobi Elliptic Function nd">jacobi_nd
</a>,
<a class=
"link" href=
"jacobi_ns.html" title=
"Jacobi Elliptic Function ns">jacobi_ns
</a>,
74 <a class=
"link" href=
"jacobi_sc.html" title=
"Jacobi Elliptic Function sc">jacobi_sc
</a>,
<a class=
"link" href=
"jacobi_sd.html" title=
"Jacobi Elliptic Function sd">jacobi_sd
</a>
75 and
<a class=
"link" href=
"jacobi_sn.html" title=
"Jacobi Elliptic Function sn">jacobi_sn
</a>.
78 They are all called as for example:
80 <pre class=
"programlisting"><span class=
"identifier">jacobi_cs
</span><span class=
"special">(
</span><span class=
"identifier">k
</span><span class=
"special">,
</span> <span class=
"identifier">u
</span><span class=
"special">);
</span>
83 Note however that these individual functions are all really thin wrappers
84 around the function
<a class=
"link" href=
"jacobi_elliptic.html" title=
"Jacobi Elliptic SN, CN and DN">jacobi_elliptic
</a>
85 which calculates the three copolar functions
<span class=
"emphasis"><em>sn
</em></span>,
<span class=
"emphasis"><em>cn
</em></span>
86 and
<span class=
"emphasis"><em>dn
</em></span> in a single function call. Thus if you need more
87 than one of these functions for a given set of arguments, it's most efficient
88 to use
<a class=
"link" href=
"jacobi_elliptic.html" title=
"Jacobi Elliptic SN, CN and DN">jacobi_elliptic
</a>.
90 <div class=
"footnotes">
91 <br><hr style=
"width:100; align:left;">
92 <div id=
"ftn.math_toolkit.jacobi.jac_over.f0" class=
"footnote"><p><a href=
"#math_toolkit.jacobi.jac_over.f0" class=
"para"><sup class=
"para">[
2]
</sup></a>
93 <a href=
"http://en.wikipedia.org/wiki/Jacobi_elliptic_functions" target=
"_top">Wikipedia:
94 Jacobi elliptic functions
</a>
96 <div id=
"ftn.math_toolkit.jacobi.jac_over.f1" class=
"footnote"><p><a href=
"#math_toolkit.jacobi.jac_over.f1" class=
"para"><sup class=
"para">[
3]
</sup></a>
97 <a href=
"http://mathworld.wolfram.com/JacobiEllipticFunctions.html" target=
"_top">Weisstein,
98 Eric W.
"Jacobi Elliptic Functions." From MathWorld - A Wolfram
101 <div id=
"ftn.math_toolkit.jacobi.jac_over.f2" class=
"footnote"><p><a href=
"#math_toolkit.jacobi.jac_over.f2" class=
"para"><sup class=
"para">[
4]
</sup></a>
102 <a href=
"http://dlmf.nist.gov/22" target=
"_top">Digital Library of Mathematical Functions:
103 Jacobian Elliptic Functions
</a>
107 <table xmlns:
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© 2006-
2010,
2012-
2014 Nikhar Agrawal,
110 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
111 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R
åde, Gautam Sewani,
112 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang
<p>
113 Distributed under the Boost Software License, Version
1.0. (See accompanying
114 file LICENSE_1_0.txt or copy at
<a href=
"http://www.boost.org/LICENSE_1_0.txt" target=
"_top">http://www.boost.org/LICENSE_1_0.txt
</a>)
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