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<a name="Struve-Functions"></a>
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<p>
Next: <a href="maxima_91.html#Hypergeometric-Functions" accesskey="n" rel="next">Hypergeometric Functions</a>, Previous: <a href="maxima_89.html#Error-Function" accesskey="p" rel="previous">Error Function</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="Struve-Functions-1"></a>
<h3 class="section">15.7 Struve Functions</h3>
<p>The Struve functions are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, <a href="https://personal.math.ubc.ca/~cbm/aands/page_496.htm">A&S Chapter 12</a> and (<a href="https://dlmf.nist.gov/11">DLMF 11</a>).
The Struve Function
\({\bf H}_{\nu}(z)\) is a particular solution
of the differential equation:
$$
z^2 {d^2 w \over dz^2} + z {dw \over dz} + (z^2-\nu^2)w =
{{4\left({1\over 2} z\right)^{\nu+1}} \over \sqrt{\pi} \Gamma\left(\nu + {1\over 2}\right)}
$$</p>
<p>which has the general soution
$$
w = aJ_{\nu}(z) + bY_{\nu}(z) + {\bf H}_{\nu}(z)
$$</p>
<a name="struve_005fh"></a><a name="Item_003a-Special_002fdeffn_002fstruve_005fh"></a><dl>
<dt><a name="index-struve_005fh"></a>Function: <strong>struve_h</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd><p>The Struve Function H of order
\(\nu\) and argument <em>z</em>:
</p>
$$
{\bf H}_{\nu}(z) = \left({z\over 2}\right)^{\nu+1}
\sum_{k=0}^{\infty} {(-1)^k\left({z\over 2}\right)^{2k} \over \Gamma\left(k + {3\over 2}\right) \Gamma\left(k + \nu + {3\over 2}\right)}
$$
<p>(<a href="https://personal.math.ubc.ca/~cbm/aands/page_496.htm">A&S eqn 12.1.3</a>) and (<a href="https://dlmf.nist.gov/11.2.E1">DLMF 11.2.E1</a>).
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>struve_h</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="maxima_85.html#besselexpand">besselexpand</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="struve_005fl"></a><a name="Item_003a-Special_002fdeffn_002fstruve_005fl"></a><dl>
<dt><a name="index-struve_005fl"></a>Function: <strong>struve_l</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd><p>The Modified Struve Function L of order
\(\nu\) and argument <em>z</em>:
$$
{\bf L}_{\nu}(z) = -ie^{-{i\nu\pi\over 2}} {\bf H}_{\nu}(iz)
$$</p>
<p>(<a href="https://personal.math.ubc.ca/~cbm/aands/page_498.htm">A&S eqn 12.2.1</a>) and (<a href="https://dlmf.nist.gov/11.2.E2">DLMF 11.2.E2</a>).
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>struve_l</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="maxima_85.html#besselexpand">besselexpand</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="Item_003a-Special_002fnode_002fHypergeometric-Functions"></a><hr>
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<p>
Next: <a href="maxima_91.html#Hypergeometric-Functions" accesskey="n" rel="next">Hypergeometric Functions</a>, Previous: <a href="maxima_89.html#Error-Function" accesskey="p" rel="previous">Error Function</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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