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<a name="Introduction-to-Elliptic-Functions-and-Integrals"></a>
<div class="header">
<p>
Next: <a href="maxima_96.html#Functions-and-Variables-for-Elliptic-Functions" accesskey="n" rel="next">Functions and Variables for Elliptic Functions</a>, Up: <a href="maxima_94.html#Elliptic-Functions" accesskey="u" rel="up">Elliptic Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-Elliptic-Functions-and-Integrals-1"></a>
<h3 class="section">16.1 Introduction to Elliptic Functions and Integrals</h3>
<p>Maxima includes support for Jacobian elliptic functions and for
complete and incomplete elliptic integrals. This includes symbolic
manipulation of these functions and numerical evaluation as well.
Definitions of these functions and many of their properties can by
found in Abramowitz and Stegun, <a href="https://personal.math.ubc.ca/~cbm/aands/page_567.htm">A&S Chapter 16</a> and
<a href="https://personal.math.ubc.ca/~cbm/aands/page_587.htm">A&S Chapter 17.</a> See also <a href="https://dlmf.nist.gov/22.2">DLMF 22.2</a>. As much as possible,
we use the definitions and relationships given in Abramowitz and Stegun.
</p>
<p>In particular, all elliptic functions and integrals use the parameter
<em>m</em> instead of the modulus <em>k</em> or the modular angle
<em>\alpha</em>. The following relationships are true:
</p>
$$
\eqalign{
m &= k^2 \cr
k &= \sin\alpha
}
$$
<p>Note that Abramowitz and Stegun uses the notation
\({\rm
sn}(u|m)\) where we use
\({\rm
sn}(u,m)\) instead. The DLMF uses modulus <em>k</em>
instead of the parameter <em>m</em>.
</p>
<p>The elliptic functions and integrals are primarily intended to support
symbolic computation. Therefore, most of derivatives of the functions
and integrals are known. However, if floating-point values are given,
a floating-point result is returned.
</p>
<p>Support for most of the other properties of elliptic functions and
integrals other than derivatives has not yet been written.
</p>
<p>Some examples of elliptic functions:
</p><div class="example">
<pre class="example">(%i1) jacobi_sn (u, m);
(%o1) jacobi_sn(u, m)
(%i2) jacobi_sn (u, 1);
(%o2) tanh(u)
(%i3) jacobi_sn (u, 0);
(%o3) sin(u)
(%i4) diff (jacobi_sn (u, m), u);
(%o4) jacobi_cn(u, m) jacobi_dn(u, m)
(%i5) diff (jacobi_sn (u, m), m);
(%o5) jacobi_cn(u, m) jacobi_dn(u, m)
elliptic_e(asin(jacobi_sn(u, m)), m)
(u - ------------------------------------)/(2 m)
1 - m
2
jacobi_cn (u, m) jacobi_sn(u, m)
+ --------------------------------
2 (1 - m)
</pre></div>
<p>Some examples of elliptic integrals:
</p><div class="example">
<pre class="example">(%i1) elliptic_f (phi, m);
(%o1) elliptic_f(phi, m)
(%i2) elliptic_f (phi, 0);
(%o2) phi
(%i3) elliptic_f (phi, 1);
phi %pi
(%o3) log(tan(--- + ---))
2 4
(%i4) elliptic_e (phi, 1);
(%o4) sin(phi)
(%i5) elliptic_e (phi, 0);
(%o5) phi
(%i6) elliptic_kc (1/2);
1
(%o6) elliptic_kc(-)
2
(%i7) makegamma (%);
2 1
gamma (-)
4
(%o7) -----------
4 sqrt(%pi)
(%i8) diff (elliptic_f (phi, m), phi);
1
(%o8) ---------------------
2
sqrt(1 - m sin (phi))
(%i9) diff (elliptic_f (phi, m), m);
elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
(%o9) (-----------------------------------------------
m
cos(phi) sin(phi)
- ---------------------)/(2 (1 - m))
2
sqrt(1 - m sin (phi))
</pre></div>
<p>Support for elliptic functions and integrals was written by Raymond
Toy. It is placed under the terms of the General Public License (GPL)
that governs the distribution of Maxima.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Elliptic-functions">Elliptic functions</a>
·</div>
<a name="Item_003a-Elliptic_002fnode_002fFunctions-and-Variables-for-Elliptic-Functions"></a><hr>
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<p>
Next: <a href="maxima_96.html#Functions-and-Variables-for-Elliptic-Functions" accesskey="n" rel="next">Functions and Variables for Elliptic Functions</a>, Up: <a href="maxima_94.html#Elliptic-Functions" accesskey="u" rel="up">Elliptic Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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