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<title>Maxima 5.47.0 Manual: Functions for Complex Numbers</title>
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<a name="Functions-for-Complex-Numbers"></a>
<div class="header">
<p>
Next: <a href="maxima_50.html#Combinatorial-Functions" accesskey="n" rel="next">Combinatorial Functions</a>, Previous: <a href="maxima_48.html#Functions-for-Numbers" accesskey="p" rel="previous">Functions for Numbers</a>, Up: <a href="maxima_47.html#Elementary-Functions" accesskey="u" rel="up">Elementary 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>
</div>
<a name="Functions-for-Complex-Numbers-1"></a>
<h3 class="section">10.2 Functions for Complex Numbers</h3>
<a name="cabs"></a><a name="Item_003a-MathFunctions_002fdeffn_002fcabs"></a><dl>
<dt><a name="index-cabs"></a>Function: <strong>cabs</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Calculates the absolute value of an expression representing a complex
number. Unlike the function <code><a href="maxima_48.html#abs">abs</a></code>, the <code>cabs</code> function always
decomposes its argument into a real and an imaginary part. If <code>x</code> and
<code>y</code> represent real variables or expressions, the <code>cabs</code> function
calculates the absolute value of <code>x + %i*y</code> as
</p>
<div class="example">
<pre class="example">(%i1) cabs (1);
(%o1) 1
</pre><pre class="example">(%i2) cabs (1 + %i);
(%o2) sqrt(2)
</pre><pre class="example">(%i3) cabs (exp (%i));
(%o3) 1
</pre><pre class="example">(%i4) cabs (exp (%pi * %i));
(%o4) 1
</pre><pre class="example">(%i5) cabs (exp (3/2 * %pi * %i));
(%o5) 1
</pre><pre class="example">(%i6) cabs (17 * exp (2 * %i));
(%o6) 17
</pre></div>
<p>If <code>cabs</code> returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
</p>
<div class="example">
<pre class="example">(%i1) cabs (a+%i*b);
2 2
(%o1) sqrt(b + a )
</pre><pre class="example">(%i2) declare(a,real,b,real);
(%o2) done
</pre><pre class="example">(%i3) cabs (a+%i*b);
2 2
(%o3) sqrt(b + a )
</pre><pre class="example">(%i4) assume(a>0,b>0);
(%o4) [a > 0, b > 0]
</pre><pre class="example">(%i5) cabs (a+%i*b);
2 2
(%o5) sqrt(b + a )
</pre></div>
<p>The <code>cabs</code> function can use known properties like symmetry properties of
complex functions to help it calculate the absolute value of an expression. If
such identities exist, they can be advertised to <code>cabs</code> using function
properties. The symmetries that <code>cabs</code> understands are: mirror symmetry,
conjugate function and complex characteristic.
</p>
<p><code>cabs</code> is a verb function and is not suitable for symbolic
calculations. For such calculations (including integration,
differentiation and taking limits of expressions containing absolute
values), use <code><a href="maxima_48.html#abs">abs</a></code>.
</p>
<p>The result of <code>cabs</code> can include the absolute value function,
<code><a href="maxima_48.html#abs">abs</a></code>, and the arc tangent, <code><a href="maxima_55.html#atan2">atan2</a></code>.
</p>
<p>When applied to a list or matrix, <code>cabs</code> automatically distributes over
the terms. Similarly, it distributes over both sides of an equation.
</p>
<p>For further ways to compute with complex numbers, see the functions
<code><a href="#rectform">rectform</a></code>, <code><a href="#realpart">realpart</a></code>, <code><a href="#imagpart">imagpart</a></code>,<!-- /@w -->
<code><a href="#carg">carg</a></code>, <code><a href="#conjugate">conjugate</a></code> and <code><a href="#polarform">polarform</a></code>.
</p>
<p>Examples:
</p>
<p>Examples with <code><a href="maxima_51.html#sqrt">sqrt</a></code> and <code><a href="maxima_55.html#sin">sin</a></code>.
</p>
<div class="example">
<pre class="example">(%i1) cabs(sqrt(1+%i*x));
2 1/4
(%o1) (x + 1)
(%i2) cabs(sin(x+%i*y));
2 2 2 2
(%o2) sqrt(cos (x) sinh (y) + sin (x) cosh (y))
</pre></div>
<p>The error function, <code><a href="maxima_89.html#erf">erf</a></code>, has mirror symmetry, which is used here in
the calculation of the absolute value with a complex argument:
</p>
<div class="example">
<pre class="example">(%i3) cabs(erf(x+%i*y));
2
(erf(%i y + x) - erf(%i y - x))
(%o3) sqrt(--------------------------------
4
2
(erf(%i y + x) + erf(%i y - x))
- --------------------------------)
4
</pre></div>
<p>Maxima knows complex identities for the Bessel functions, which allow
it to compute the absolute value for complex arguments. Here is an
example for <code><a href="maxima_85.html#bessel_005fj">bessel_j</a></code>.
</p>
<div class="example">
<pre class="example">(%i4) cabs(bessel_j(1,%i));
(%o4) abs(bessel_j(1, %i))
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·</div></dd></dl>
<a name="carg"></a><a name="Item_003a-MathFunctions_002fdeffn_002fcarg"></a><dl>
<dt><a name="index-carg"></a>Function: <strong>carg</strong> <em>(<var>z</var>)</em></dt>
<dd>
<p>Returns the complex argument of <var>z</var>. The complex argument is an angle
<code>theta</code> in <code>(-%pi, %pi]</code> such that <code>r exp (theta %i) = <var>z</var></code>
where <code>r</code> is the magnitude of <var>z</var>.
</p>
<p><code>carg</code> is a computational function, not a simplifying function.
</p>
<p>See also <code><a href="maxima_48.html#abs">abs</a></code> (complex magnitude), <code><a href="#polarform">polarform</a></code>,<!-- /@w -->
<code><a href="#rectform">rectform</a></code>, <code><a href="#realpart">realpart</a></code>, and <code><a href="#imagpart">imagpart</a></code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) carg (1);
(%o1) 0
</pre><pre class="example">(%i2) carg (1 + %i);
%pi
(%o2) ---
4
</pre><pre class="example">(%i3) carg (exp (%i));
sin(1)
(%o3) atan(------)
cos(1)
</pre><pre class="example">(%i4) carg (exp (%pi * %i));
(%o4) %pi
</pre><pre class="example">(%i5) carg (exp (3/2 * %pi * %i));
%pi
(%o5) - ---
2
</pre><pre class="example">(%i6) carg (17 * exp (2 * %i));
sin(2)
(%o6) atan(------) + %pi
cos(2)
</pre></div>
<p>If <code>carg</code> returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
</p>
<div class="example">
<pre class="example">(%i1) carg (a+%i*b);
(%o1) atan2(b, a)
</pre><pre class="example">(%i2) declare(a,real,b,real);
(%o2) done
</pre><pre class="example">(%i3) carg (a+%i*b);
(%o3) atan2(b, a)
</pre><pre class="example">(%i4) assume(a>0,b>0);
(%o4) [a > 0, b > 0]
</pre><pre class="example">(%i5) carg (a+%i*b);
b
(%o5) atan(-)
a
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·</div></dd></dl>
<a name="conjugate"></a><a name="Item_003a-MathFunctions_002fdeffn_002fconjugate"></a><dl>
<dt><a name="index-conjugate"></a>Function: <strong>conjugate</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Returns the complex conjugate of <var>x</var>.
</p>
<div class="example">
<pre class="example">(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary);
(%o1) done
</pre><pre class="example">(%i2) conjugate (aa + bb*%i);
(%o2) aa - %i bb
</pre><pre class="example">(%i3) conjugate (cc);
(%o3) conjugate(cc)
</pre><pre class="example">(%i4) conjugate (ii);
(%o4) - ii
</pre><pre class="example">(%i5) conjugate (xx + yy);
(%o5) yy + xx
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·</div></dd></dl>
<a name="imagpart"></a><a name="Item_003a-MathFunctions_002fdeffn_002fimagpart"></a><dl>
<dt><a name="index-imagpart"></a>Function: <strong>imagpart</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Returns the imaginary part of the expression <var>expr</var>.
</p>
<p><code>imagpart</code> is a computational function, not a simplifying function.
</p>
<p>See also <code><a href="maxima_48.html#abs">abs</a></code>, <code><a href="#carg">carg</a></code>, <code><a href="#polarform">polarform</a></code>,<!-- /@w -->
<code><a href="#rectform">rectform</a></code>, and <code><a href="#realpart">realpart</a></code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) imagpart (a+b*%i);
(%o1) b
</pre><pre class="example">(%i2) imagpart (1+sqrt(2)*%i);
(%o2) sqrt(2)
</pre><pre class="example">(%i3) imagpart (1);
(%o3) 0
</pre><pre class="example">(%i4) imagpart (sqrt(2)*%i);
(%o4) sqrt(2)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·</div></dd></dl>
<a name="polarform"></a><a name="Item_003a-MathFunctions_002fdeffn_002fpolarform"></a><dl>
<dt><a name="index-polarform"></a>Function: <strong>polarform</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Returns an expression <code>r %e^(%i theta)</code> equivalent to <var>expr</var>,
such that <code>r</code> and <code>theta</code> are purely real.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) polarform(a+b*%i);
2 2 %i atan2(b, a)
(%o1) sqrt(b + a ) %e
</pre><pre class="example">(%i2) polarform(1+%i);
%i %pi
------
4
(%o2) sqrt(2) %e
</pre><pre class="example">(%i3) polarform(1+2*%i);
%i atan(2)
(%o3) sqrt(5) %e
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·<a href="maxima_424.html#Category_003a-Exponential-and-logarithm-functions">Exponential and logarithm functions</a>
·</div></dd></dl>
<a name="realpart"></a><a name="Item_003a-MathFunctions_002fdeffn_002frealpart"></a><dl>
<dt><a name="index-realpart"></a>Function: <strong>realpart</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Returns the real part of <var>expr</var>. <code>realpart</code> and <code><a href="#imagpart">imagpart</a></code> will
work on expressions involving trigonometric and hyperbolic functions,
as well as square root, logarithm, and exponentiation.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) realpart (a+b*%i);
(%o1) a
</pre><pre class="example">(%i2) realpart (1+sqrt(2)*%i);
(%o2) 1
</pre><pre class="example">(%i3) realpart (sqrt(2)*%i);
(%o3) 0
</pre><pre class="example">(%i4) realpart (1);
(%o4) 1
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·</div></dd></dl>
<a name="rectform"></a><a name="Item_003a-MathFunctions_002fdeffn_002frectform"></a><dl>
<dt><a name="index-rectform"></a>Function: <strong>rectform</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Returns an expression <code>a + b %i</code> equivalent to <var>expr</var>,
such that <var>a</var> and <var>b</var> are purely real.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) rectform(sqrt(2)*%e^(%i*%pi/4));
(%o1) %i + 1
</pre><pre class="example">(%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a)));
(%o2) %i b + a
</pre><pre class="example">(%i3) rectform(sqrt(5)*%e^(%i*atan(2)));
(%o3) 2 %i + 1
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
·</div></dd></dl>
<a name="Item_003a-MathFunctions_002fnode_002fCombinatorial-Functions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_50.html#Combinatorial-Functions" accesskey="n" rel="next">Combinatorial Functions</a>, Previous: <a href="maxima_48.html#Functions-for-Numbers" accesskey="p" rel="previous">Functions for Numbers</a>, Up: <a href="maxima_47.html#Elementary-Functions" accesskey="u" rel="up">Elementary 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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