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<p>
Next: <a href="maxima_58.html#Gamma-and-Factorial-Functions" accesskey="n" rel="next">Gamma and Factorial Functions</a>, Previous: <a href="maxima_56.html#Bessel-Functions" accesskey="p" rel="previous">Bessel Functions</a>, Up: <a href="maxima_54.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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<a name="Airy-Functions-1"></a>
<h3 class="section">14.3 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&amp;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 &lt; 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&amp;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>




</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>




</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&amp;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>




</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>




</dd></dl>

<a name="Item_003a-Special_002fnode_002fGamma-and-Factorial-Functions"></a><hr>
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Next: <a href="maxima_58.html#Gamma-and-Factorial-Functions" accesskey="n" rel="next">Gamma and Factorial Functions</a>, Previous: <a href="maxima_56.html#Bessel-Functions" accesskey="p" rel="previous">Bessel Functions</a>, Up: <a href="maxima_54.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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