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<a name="Airy-Functions"></a>
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<p>
Next: <a href="maxima_87.html#Gamma-and-Factorial-Functions" accesskey="n" rel="next">Gamma and Factorial Functions</a>, Previous: <a href="maxima_85.html#Bessel-Functions" accesskey="p" rel="previous">Bessel Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special 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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<a name="Airy-Functions-1"></a>
<h3 class="section">15.3 Airy Functions</h3>
<p>The Airy functions
\({\rm Ai}(x)\) and
\({\rm Bi}(x)\) are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, <a href="https://personal.math.ubc.ca/~cbm/aands/page_446.htm">A&S Section 10.4</a> and <a href="https://dlmf.nist.gov/9">DLMF 9</a>.
</p>
<p>The Airy differential equation is:
</p>
$$
{d^2 y\over dx^2} - xy = 0
$$
<p>The numerically satisfactory pair of solutions (<a href="https://dlmf.nist.gov/9.2#T1">DLMF 9.2#T1</a>) on the real line are
\(y = {\rm Ai}(x)\) and
\(y = {\rm Bi}(x).\)</p>
<p>These two solutions are oscillatory for <em>x < 0</em>.
\({\rm Ai}(x)\) is
the solution subject to the condition that
\(y\rightarrow 0\) as
\(x\rightarrow +\infty,\) and
\({\rm Bi}(x)\) is
the second solution with the
same amplitude as
\({\rm Ai}(x)\) as
\(x\rightarrow-\infty\) which differs in phase
by
\(\pi/2.\) Also,
\({\rm Bi}(x)\) is unbounded
as
\(x\rightarrow +\infty.\)</p>
<p>If the argument <em>x</em> is a real or complex floating point
number, the numerical value of the function is returned.
</p>
<a name="airy_005fai"></a><a name="Item_003a-Special_002fdeffn_002fairy_005fai"></a><dl>
<dt><a name="index-airy_005fai"></a>Function: <strong>airy_ai</strong> <em>(<var>x</var>)</em></dt>
<dd><p>The Airy function
\({\rm Ai}(x).\) See <a href="https://personal.math.ubc.ca/~cbm/aands/page_446.htm">A&S eqn 10.4.2</a> and <a href="https://dlmf.nist.gov/9">DLMF 9</a>.
</p>
<p>See also <code><a href="#airy_005fbi">airy_bi</a></code>, <code><a href="#airy_005fdai">airy_dai</a></code>, and <code><a href="#airy_005fdbi">airy_dbi</a></code>.
</p>
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Categories:<a href="maxima_424.html#Category_003a-Airy-functions">Airy functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="airy_005fdai"></a><a name="Item_003a-Special_002fdeffn_002fairy_005fdai"></a><dl>
<dt><a name="index-airy_005fdai"></a>Function: <strong>airy_dai</strong> <em>(<var>x</var>)</em></dt>
<dd><p>The derivative of the Airy function
\({\rm Ai}(x)\):
</p>
$$
{\rm airy\_dai}(x) = {d\over dx}{\rm Ai}(x)
$$
<p>See <code><a href="#airy_005fai">airy_ai</a></code>.
</p>
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Categories:<a href="maxima_424.html#Category_003a-Airy-functions">Airy functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="airy_005fbi"></a><a name="Item_003a-Special_002fdeffn_002fairy_005fbi"></a><dl>
<dt><a name="index-airy_005fbi"></a>Function: <strong>airy_bi</strong> <em>(<var>x</var>)</em></dt>
<dd><p>The Airy function
\({\rm Bi}(x)\). See <a href="https://personal.math.ubc.ca/~cbm/aands/page_446.htm">A&S eqn 10.4.3</a> and <a href="https://dlmf.nist.gov/9">DLMF 9</a>.
</p>
<p>See <code><a href="#airy_005fai">airy_ai</a></code>, and <code><a href="#airy_005fdbi">airy_dbi</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Airy-functions">Airy functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="airy_005fdbi"></a><a name="Item_003a-Special_002fdeffn_002fairy_005fdbi"></a><dl>
<dt><a name="index-airy_005fdbi"></a>Function: <strong>airy_dbi</strong> <em>(<var>x</var>)</em></dt>
<dd><p>The derivative of the Airy function
\({\rm Bi}(x)\):
</p>
$$
{\rm airy\_dbi}(x) = {d\over dx}{\rm Bi}(x)
$$
<p>See <code><a href="#airy_005fai">airy_ai</a></code>, and <code><a href="#airy_005fbi">airy_bi</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Airy-functions">Airy functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="Item_003a-Special_002fnode_002fGamma-and-Factorial-Functions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_87.html#Gamma-and-Factorial-Functions" accesskey="n" rel="next">Gamma and Factorial Functions</a>, Previous: <a href="maxima_85.html#Bessel-Functions" accesskey="p" rel="previous">Bessel Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special 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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