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<a name="Functions-and-Variables-for-Special-Functions"></a>
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<p>
Previous: <a href="maxima_63.html#Parabolic-Cylinder-Functions" accesskey="p" rel="previous">Parabolic Cylinder Functions</a>, Up: <a href="maxima_54.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_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="Functions-and-Variables-for-Special-Functions-1"></a>
<h3 class="section">14.10 Functions and Variables for Special Functions</h3>
<a name="Item_003a-Special_002fdeffn_002flambert_005fw"></a><dl>
<dt><a name="index-lambert_005fw"></a>Function: <strong>lambert_w</strong> <em>(<var>z</var>)</em></dt>
<dd><p>The principal branch of Lambert’s W function W(z) (<a href="https://dlmf.nist.gov/4.13">DLMF 4.13</a>), the solution of
$$
z = W(z)e^{W(z)}
$$</p>
</dd></dl>
<a name="Item_003a-Special_002fdeffn_002fgeneralized_005flambert_005fw"></a><dl>
<dt><a name="index-generalized_005flambert_005fw"></a>Function: <strong>generalized_lambert_w</strong> <em>(<var>k</var>, <var>z</var>)</em></dt>
<dd><p>The <var>k</var>-th branch of Lambert’s W function W(z) (<a href="https://dlmf.nist.gov/4.13">DLMF 4.13</a>), the solution
of
\(z=W(z)e^{W(z)}\).
</p>
<p>The principal branch, denoted
\(W_p(z)\) in DLMF, is <code>lambert_w(z) = generalized_lambert_w(0,z)</code>.
</p>
<p>The other branch with real values, denoted
\(W_m(z)\) in DLMF, is <code>generalized_lambert_w(-1,z)</code>.
</p>
</dd></dl>
<a name="kbateman"></a><a name="Item_003a-Special_002fdeffn_002fkbateman"></a><dl>
<dt><a name="index-kbateman"></a>Function: <strong>kbateman</strong> <em>[<var>v</var>] (<var>x</var>)</em></dt>
<dd><p>The Bateman k function
</p>
$$
k_v(x)
= \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \cos(x \tan\theta-v\theta)d\theta
$$
<p>It is a special case of the confluent hypergeometric function. Maxima can
calculate the Laplace transform of <code>kbateman</code> using <code><a href="maxima_75.html#laplace">laplace</a></code>
or <code><a href="maxima_75.html#specint">specint</a></code>, but has no other knowledge of this function.
</p>
</dd></dl>
<a name="Item_003a-Special_002fdeffn_002fnzeta"></a><dl>
<dt><a name="index-nzeta"></a>Function: <strong>nzeta</strong> <em>(<var>z</var>)</em></dt>
<dd><p>The Plasma Dispersion Function
$$
{\rm nzeta}(z) = i\sqrt{\pi}e^{-z^2}(1-{\rm erf}(-iz))
$$</p>
</dd></dl>
<a name="Item_003a-Special_002fdeffn_002fnzetar"></a><dl>
<dt><a name="index-nzetar"></a>Function: <strong>nzetar</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Returns <code>realpart(nzeta(z))</code>.
</p>
</dd></dl>
<a name="Item_003a-Special_002fdeffn_002fnzetai"></a><dl>
<dt><a name="index-nzetai"></a>Function: <strong>nzetai</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Returns <code>imagpart(nzeta(z))</code>.
</p>
</dd></dl>
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