1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
|
<!DOCTYPE html>
<html>
<!-- Created by GNU Texinfo 7.1.1, https://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Complex Arithmetic (GNU Octave (version 10.3.0))</title>
<meta name="description" content="Complex Arithmetic (GNU Octave (version 10.3.0))">
<meta name="keywords" content="Complex Arithmetic (GNU Octave (version 10.3.0))">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta name="viewport" content="width=device-width,initial-scale=1">
<link href="index.html" rel="start" title="Top">
<link href="Concept-Index.html" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Arithmetic.html" rel="up" title="Arithmetic">
<link href="Trigonometry.html" rel="next" title="Trigonometry">
<link href="Exponents-and-Logarithms.html" rel="prev" title="Exponents and Logarithms">
<style type="text/css">
<!--
a.copiable-link {visibility: hidden; text-decoration: none; line-height: 0em}
div.example {margin-left: 3.2em}
span:hover a.copiable-link {visibility: visible}
strong.def-name {font-family: monospace; font-weight: bold; font-size: larger}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">
</head>
<body lang="en">
<div class="section-level-extent" id="Complex-Arithmetic">
<div class="nav-panel">
<p>
Next: <a href="Trigonometry.html" accesskey="n" rel="next">Trigonometry</a>, Previous: <a href="Exponents-and-Logarithms.html" accesskey="p" rel="prev">Exponents and Logarithms</a>, Up: <a href="Arithmetic.html" 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" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<h3 class="section" id="Complex-Arithmetic-1"><span>17.2 Complex Arithmetic<a class="copiable-link" href="#Complex-Arithmetic-1"> ¶</a></span></h3>
<p>In the descriptions of the following functions,
<var class="var">z</var> is the complex number <var class="var">x</var> + <var class="var">i</var><var class="var">y</var>, where <var class="var">i</var> is
defined as <code class="code">sqrt (-1)</code>.
</p>
<a class="anchor" id="XREFabs"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-abs"><span><code class="def-type"><var class="var">z</var> =</code> <strong class="def-name">abs</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-abs"> ¶</a></span></dt>
<dd><p>Compute the magnitude of <var class="var">x</var>.
</p>
<p>The magnitude is defined as
|<var class="var">z</var>| = <code class="code">sqrt (x^2 + y^2)</code>.
</p>
<p>For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">abs (3 + 4i)
⇒ 5
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFarg">arg</a>.
</p></dd></dl>
<a class="anchor" id="XREFarg"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-arg"><span><code class="def-type"><var class="var">theta</var> =</code> <strong class="def-name">arg</strong> <code class="def-code-arguments">(<var class="var">z</var>)</code><a class="copiable-link" href="#index-arg"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-angle"><span><code class="def-type"><var class="var">theta</var> =</code> <strong class="def-name">angle</strong> <code class="def-code-arguments">(<var class="var">z</var>)</code><a class="copiable-link" href="#index-angle"> ¶</a></span></dt>
<dd><p>Compute the argument, i.e., angle of <var class="var">z</var>.
</p>
<p>This is defined as,
<var class="var">theta</var> = <code class="code">atan2 (<var class="var">y</var>, <var class="var">x</var>)</code>,
in radians.
</p>
<p>For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">arg (3 + 4i)
⇒ 0.92730
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFabs">abs</a>.
</p></dd></dl>
<a class="anchor" id="XREFconj"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-conj"><span><code class="def-type"><var class="var">zc</var> =</code> <strong class="def-name">conj</strong> <code class="def-code-arguments">(<var class="var">z</var>)</code><a class="copiable-link" href="#index-conj"> ¶</a></span></dt>
<dd><p>Return the complex conjugate of <var class="var">z</var>.
</p>
<p>The complex conjugate is defined as
<code class="code">conj (<var class="var">z</var>)</code> = <var class="var">x</var> - <var class="var">i</var><var class="var">y</var>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFreal">real</a>, <a class="ref" href="#XREFimag">imag</a>.
</p></dd></dl>
<a class="anchor" id="XREFcplxpair"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-cplxpair"><span><code class="def-type"><var class="var">zsort</var> =</code> <strong class="def-name">cplxpair</strong> <code class="def-code-arguments">(<var class="var">z</var>)</code><a class="copiable-link" href="#index-cplxpair"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-cplxpair-1"><span><code class="def-type"><var class="var">zsort</var> =</code> <strong class="def-name">cplxpair</strong> <code class="def-code-arguments">(<var class="var">z</var>, <var class="var">tol</var>)</code><a class="copiable-link" href="#index-cplxpair-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-cplxpair-2"><span><code class="def-type"><var class="var">zsort</var> =</code> <strong class="def-name">cplxpair</strong> <code class="def-code-arguments">(<var class="var">z</var>, <var class="var">tol</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-cplxpair-2"> ¶</a></span></dt>
<dd><p>Sort the numbers <var class="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 class="code">abs (imag (<var class="var">z</var>)) / abs (<var class="var">z</var>) < <var class="var">tol</var></code>) are placed after
the complex pairs.
</p>
<p><var class="var">tol</var> is a weighting factor in the range [0, 1) which determines the
tolerance of the matching. The default value is <code class="code">100 * eps</code> and the
resulting tolerance for a given complex pair is
<code class="code"><var class="var">tol</var> * abs (<var class="var">z</var>(i)))</code>.
</p>
<p>By default the complex pairs are sorted along the first non-singleton
dimension of <var class="var">z</var>. If <var class="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 class="var">tol</var>).
Note that there is no defined order for pairs with identical real parts but
differing imaginary parts.
</p>
<div class="example smallexample">
<pre class="example-preformatted">cplxpair (exp (2i*pi*[0:4]'/5)) == exp (2i*pi*[3; 2; 4; 1; 0]/5)
</pre></div>
</dd></dl>
<a class="anchor" id="XREFimag"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-imag"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">imag</strong> <code class="def-code-arguments">(<var class="var">z</var>)</code><a class="copiable-link" href="#index-imag"> ¶</a></span></dt>
<dd><p>Return the imaginary part of <var class="var">z</var> as a real number.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFreal">real</a>, <a class="ref" href="#XREFconj">conj</a>.
</p></dd></dl>
<a class="anchor" id="XREFreal"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-real"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">real</strong> <code class="def-code-arguments">(<var class="var">z</var>)</code><a class="copiable-link" href="#index-real"> ¶</a></span></dt>
<dd><p>Return the real part of <var class="var">z</var>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFimag">imag</a>, <a class="ref" href="#XREFconj">conj</a>.
</p></dd></dl>
</div>
<hr>
<div class="nav-panel">
<p>
Next: <a href="Trigonometry.html">Trigonometry</a>, Previous: <a href="Exponents-and-Logarithms.html">Exponents and Logarithms</a>, Up: <a href="Arithmetic.html">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
