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<a name="Complex-Arithmetic"></a>
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<a name="Complex-Arithmetic-1"></a>
<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>.
</p>
<a name="XREFabs"></a><dl>
<dt><a name="index-abs"></a><em></em> <strong>abs</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Compute the magnitude of <var>z</var>.
</p>
<p>The magnitude is defined as
|<var>z</var>| = <code>sqrt (x^2 + y^2)</code>.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">abs (3 + 4i)
⇒ 5
</pre></div>
<p><strong>See also:</strong> <a href="#XREFarg">arg</a>.
</p></dd></dl>
<a name="XREFarg"></a><dl>
<dt><a name="index-arg"></a><em></em> <strong>arg</strong> <em>(<var>z</var>)</em></dt>
<dt><a name="index-angle"></a><em></em> <strong>angle</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Compute the argument, i.e., angle of <var>z</var>.
</p>
<p>This is defined as,
<var>theta</var> = <code>atan2 (<var>y</var>, <var>x</var>)</code>,
in radians.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">arg (3 + 4i)
⇒ 0.92730
</pre></div>
<p><strong>See also:</strong> <a href="#XREFabs">abs</a>.
</p></dd></dl>
<a name="XREFconj"></a><dl>
<dt><a name="index-conj"></a><em></em> <strong>conj</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Return the complex conjugate of <var>z</var>.
</p>
<p>The complex conjugate is defined as
<code>conj (<var>z</var>)</code> = <var>x</var> - <var>i</var><var>y</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFreal">real</a>, <a href="#XREFimag">imag</a>.
</p></dd></dl>
<a name="XREFcplxpair"></a><dl>
<dt><a name="index-cplxpair"></a><em></em> <strong>cplxpair</strong> <em>(<var>z</var>)</em></dt>
<dt><a name="index-cplxpair-1"></a><em></em> <strong>cplxpair</strong> <em>(<var>z</var>, <var>tol</var>)</em></dt>
<dt><a name="index-cplxpair-2"></a><em></em> <strong>cplxpair</strong> <em>(<var>z</var>, <var>tol</var>, <var>dim</var>)</em></dt>
<dd><p>Sort the numbers <var>z</var> into complex conjugate pairs ordered by increasing
real part.
</p>
<p>The negative imaginary complex numbers are placed first within each pair.
All real numbers (those with
<code>abs (imag (<var>z</var>)) / abs (<var>z</var>) < <var>tol</var></code>) are placed after
the complex pairs.
</p>
<p><var>tol</var> is a weighting factor in the range [0, 1) which determines the
tolerance of the matching. The default value is <code>100 * eps</code> and the
resulting tolerance for a given complex pair is
<code><var>tol</var> * abs (<var>z</var>(i)))</code>.
</p>
<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>
<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.
</p>
<div class="smallexample">
<pre class="smallexample">cplxpair (exp (2i*pi*[0:4]'/5)) == exp (2i*pi*[3; 2; 4; 1; 0]/5)
</pre></div>
</dd></dl>
<a name="XREFimag"></a><dl>
<dt><a name="index-imag"></a><em></em> <strong>imag</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Return the imaginary part of <var>z</var> as a real number.
</p>
<p><strong>See also:</strong> <a href="#XREFreal">real</a>, <a href="#XREFconj">conj</a>.
</p></dd></dl>
<a name="XREFreal"></a><dl>
<dt><a name="index-real"></a><em></em> <strong>real</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Return the real part of <var>z</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFimag">imag</a>, <a href="#XREFconj">conj</a>.
</p></dd></dl>
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Next: <a href="Trigonometry.html#Trigonometry" accesskey="n" rel="next">Trigonometry</a>, Previous: <a href="Exponents-and-Logarithms.html#Exponents-and-Logarithms" accesskey="p" rel="prev">Exponents and Logarithms</a>, Up: <a href="Arithmetic.html#Arithmetic" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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