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<h3 class="section">17.2 Complex Arithmetic</h3>
<p>In the descriptions of the following functions,
<var>z</var> is the complex number <var>x</var> + <var>i</var><var>y</var>, where <var>i</var> is
defined as <code>sqrt (-1)</code>.
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<p><a name="doc_002dabs"></a>
<div class="defun">
— Mapping Function: <b>abs</b> (<var>z</var>)<var><a name="index-abs-1376"></a></var><br>
<blockquote><p>Compute the magnitude of <var>z</var>, defined as
|<var>z</var>| = <code>sqrt (x^2 + y^2)</code>.
<p>For example,
<pre class="example"> abs (3 + 4i)
⇒ 5
</pre>
</blockquote></div>
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<p><a name="doc_002darg"></a>
<div class="defun">
— Mapping Function: <b>arg</b> (<var>z</var>)<var><a name="index-arg-1377"></a></var><br>
— Mapping Function: <b>angle</b> (<var>z</var>)<var><a name="index-angle-1378"></a></var><br>
<blockquote><p>Compute the argument of <var>z</var>, defined as,
<var>theta</var> = <code>atan2 (</code><var>y</var><code>, </code><var>x</var><code>)</code>,
in radians.
<p>For example,
<pre class="example"> arg (3 + 4i)
⇒ 0.92730
</pre>
</blockquote></div>
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<p><a name="doc_002dconj"></a>
<div class="defun">
— Mapping Function: <b>conj</b> (<var>z</var>)<var><a name="index-conj-1379"></a></var><br>
<blockquote><p>Return the complex conjugate of <var>z</var>, defined as
<code>conj (</code><var>z</var><code>)</code> = <var>x</var> - <var>i</var><var>y</var>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dreal.html#doc_002dreal">real</a>, <a href="doc_002dimag.html#doc_002dimag">imag</a>.
</p></blockquote></div>
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<p><a name="doc_002dcplxpair"></a>
<div class="defun">
— Function File: <b>cplxpair</b> (<var>z</var>)<var><a name="index-cplxpair-1380"></a></var><br>
— Function File: <b>cplxpair</b> (<var>z, tol</var>)<var><a name="index-cplxpair-1381"></a></var><br>
— Function File: <b>cplxpair</b> (<var>z, tol, dim</var>)<var><a name="index-cplxpair-1382"></a></var><br>
<blockquote><p>Sort the numbers <var>z</var> into complex conjugate pairs ordered by
increasing real part. Place the negative imaginary complex number
first within each pair. Place all the real numbers (those with
<code>abs (imag (</code><var>z</var><code>) / </code><var>z</var><code>) < </code><var>tol</var><code>)</code>) after the
complex pairs.
<p>If <var>tol</var> is unspecified the default value is 100*<code>eps</code>.
<p>By default the complex pairs are sorted along the first non-singleton
dimension of <var>z</var>. If <var>dim</var> is specified, then the complex
pairs are sorted along this dimension.
<p>Signal an error if some complex numbers could not be paired. Signal an
error if all complex numbers are not exact conjugates (to within
<var>tol</var>). Note that there is no defined order for pairs with identical
real parts but differing imaginary parts.
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<pre class="smallexample"> cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5)
</pre>
</blockquote></div>
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<p><a name="doc_002dimag"></a>
<div class="defun">
— Mapping Function: <b>imag</b> (<var>z</var>)<var><a name="index-imag-1383"></a></var><br>
<blockquote><p>Return the imaginary part of <var>z</var> as a real number.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dreal.html#doc_002dreal">real</a>, <a href="doc_002dconj.html#doc_002dconj">conj</a>.
</p></blockquote></div>
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<p><a name="doc_002dreal"></a>
<div class="defun">
— Mapping Function: <b>real</b> (<var>z</var>)<var><a name="index-real-1384"></a></var><br>
<blockquote><p>Return the real part of <var>z</var>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dimag.html#doc_002dimag">imag</a>, <a href="doc_002dconj.html#doc_002dconj">conj</a>.
</p></blockquote></div>
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