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<a name="Package-functs"></a>
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<p>
Next: <a href="maxima_361.html#Package-ineq" accesskey="n" rel="next">Package ineq</a>, Previous: <a href="maxima_359.html#Package-facexp" accesskey="p" rel="previous">Package facexp</a>, Up: <a href="maxima_356.html#simplification_002dpkg" accesskey="u" rel="up">simplification-pkg</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="Package-functs-1"></a>
<h3 class="section">87.4 Package functs</h3>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Share-packages">Share packages</a>
·<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·</div>
<a name="rempart"></a><a name="Item_003a-simplifications_002fdeffn_002frempart"></a><dl>
<dt><a name="index-rempart"></a>Function: <strong>rempart</strong> <em>(<var>expr</var>, <var>n</var>)</em></dt>
<dd><p>Removes part <var>n</var> from the expression <var>expr</var>.
</p>
<p>If <var>n</var> is a list of the form <code>[<var>l</var>, <var>m</var>]</code>
then parts <var>l</var> thru <var>m</var> are removed.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Expressions">Expressions</a>
·</div>
</dd></dl>
<a name="wronskian"></a><a name="Item_003a-simplifications_002fdeffn_002fwronskian"></a><dl>
<dt><a name="index-wronskian"></a>Function: <strong>wronskian</strong> <em>([<var>f_1</var>, ..., <var>f_n</var>], <var>x</var>)</em></dt>
<dd><p>Returns the Wronskian matrix of the list of expressions [<var>f_1</var>, ..., <var>f_n</var>] in the variable <var>x</var>.
The determinant of the Wronskian matrix is the Wronskian determinant of the list of expressions.
</p>
<p>To use <code>wronskian</code>, first <code>load("functs")</code>. Example:
</p>
<div class="example">
<pre class="example">(%i1) load ("functs")$
</pre><pre class="example">(%i2) wronskian([f(x), g(x)],x);
[ f(x) g(x) ]
[ ]
(%o2) [ d d ]
[ -- (f(x)) -- (g(x)) ]
[ dx dx ]
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Differential-calculus">Differential calculus</a>
·</div>
</dd></dl>
<a name="tracematrix"></a><a name="Item_003a-simplifications_002fdeffn_002ftracematrix"></a><dl>
<dt><a name="index-tracematrix"></a>Function: <strong>tracematrix</strong> <em>(<var>M</var>)</em></dt>
<dd><p>Returns the trace (sum of the diagonal elements) of matrix <var>M</var>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Matrices">Matrices</a>
·</div>
</dd></dl>
<a name="Item_003a-simplifications_002fdeffn_002frational"></a><dl>
<dt><a name="index-rational"></a>Function: <strong>rational</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Multiplies numerator and denominator of <var>z</var> by the complex conjugate of denominator,
thus rationalizing the denominator.
Returns canonical rational expression (CRE) form if given one, else returns general form.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Expressions">Expressions</a>
·</div>
</dd></dl>
<a name="nonzeroandfreeof"></a><a name="Item_003a-simplifications_002fdeffn_002fnonzeroandfreeof"></a><dl>
<dt><a name="index-nonzeroandfreeof"></a>Function: <strong>nonzeroandfreeof</strong> <em>(<var>x</var>, <var>expr</var>)</em></dt>
<dd><p>Returns <code>true</code> if <var>expr</var> is nonzero and <code>freeof (<var>x</var>, <var>expr</var>)</code> returns <code>true</code>.
Returns <code>false</code> otherwise.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Expressions">Expressions</a>
·</div>
</dd></dl>
<a name="Item_003a-simplifications_002fdeffn_002flinear"></a><dl>
<dt><a name="index-linear"></a>Function: <strong>linear</strong> <em>(<var>expr</var>, <var>x</var>)</em></dt>
<dd><p>When <var>expr</var> is an expression of the form <code><var>a</var>*<var>x</var> + <var>b</var></code>
where <var>a</var> is nonzero, and <var>a</var> and <var>b</var> are free of <var>x</var>,
<code>linear</code> returns a list of three equations, one for each of the three formal
variables <var>b</var>, <var>a</var>, and <var>x</var>. Otherwise, <code>linear</code> returns <code>false</code>.
</p>
<p><code>load("antid")</code> loads this function.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) load ("antid");
(%o1) /maxima/share/integration/antid.mac
</pre><pre class="example">(%i2) linear ((1 - w)*(1 - x)*z, z);
(%o2) [bargumentb = 0, aargumenta = (w - 1) x - w + 1,
xargumentx = z]
</pre><pre class="example">(%i3) linear (cos(u - v) + cos(u + v), u);
(%o3) false
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-antid">Package antid</a>
·<a href="maxima_424.html#Category_003a-Expressions">Expressions</a>
·</div>
</dd></dl>
<a name="gcdivide"></a><a name="Item_003a-simplifications_002fdeffn_002fgcdivide"></a><dl>
<dt><a name="index-gcdivide"></a>Function: <strong>gcdivide</strong> <em>(<var>p</var>, <var>q</var>)</em></dt>
<dd>
<p>When the option variable <code>takegcd</code> is <code>true</code> which is the default,
<code>gcdivide</code> divides the polynomials <var>p</var> and <var>q</var> by their greatest
common divisor and returns the ratio of the results. <code>gcdivde</code> calls the
function <code><a href="maxima_80.html#ezgcd">ezgcd</a></code> to divide the polynomials by the greatest common divisor.
</p>
<p>When <code>takegcd</code> is <code>false</code>, <code>gcdivide</code> returns the ratio
<code><var>p</var>/<var>q</var></code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<p>See also <code><a href="maxima_80.html#ezgcd">ezgcd</a></code>, <code><a href="maxima_80.html#gcd">gcd</a></code>, <code><a href="maxima_80.html#gcdex">gcdex</a></code>, and
<code><a href="maxima_290.html#poly_005fgcd">poly_gcd</a></code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) load("functs")$
(%i2) p1:6*x^3+19*x^2+19*x+6;
3 2
(%o2) 6 x + 19 x + 19 x + 6
(%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
5 4 3 2
(%o3) 6 x + 13 x + 12 x + 13 x + 6 x
(%i4) gcdivide(p1, p2);
x + 1
(%o4) ------
3
x + x
(%i5) takegcd:false;
(%o5) false
(%i6) gcdivide(p1, p2);
3 2
6 x + 19 x + 19 x + 6
(%o6) ----------------------------------
5 4 3 2
6 x + 13 x + 12 x + 13 x + 6 x
(%i7) ratsimp(%);
x + 1
(%o7) ------
3
x + x
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Polynomials">Polynomials</a>
·</div></dd></dl>
<a name="arithmetic"></a><a name="Item_003a-simplifications_002fdeffn_002farithmetic"></a><dl>
<dt><a name="index-arithmetic"></a>Function: <strong>arithmetic</strong> <em>(<var>a</var>, <var>d</var>, <var>n</var>)</em></dt>
<dd><p>Returns the <var>n</var>-th term of the arithmetic series
<code><var>a</var>, <var>a</var> + <var>d</var>, <var>a</var> + 2*<var>d</var>, ..., <var>a</var> + (<var>n</var> - 1)*<var>d</var></code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Sums-and-products">Sums and products</a>
·</div>
</dd></dl>
<a name="geometric"></a><a name="Item_003a-simplifications_002fdeffn_002fgeometric"></a><dl>
<dt><a name="index-geometric"></a>Function: <strong>geometric</strong> <em>(<var>a</var>, <var>r</var>, <var>n</var>)</em></dt>
<dd><p>Returns the <var>n</var>-th term of the geometric series
<code><var>a</var>, <var>a</var>*<var>r</var>, <var>a</var>*<var>r</var>^2, ..., <var>a</var>*<var>r</var>^(<var>n</var> - 1)</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Sums-and-products">Sums and products</a>
·</div>
</dd></dl>
<a name="harmonic"></a><a name="Item_003a-simplifications_002fdeffn_002fharmonic"></a><dl>
<dt><a name="index-harmonic"></a>Function: <strong>harmonic</strong> <em>(<var>a</var>, <var>b</var>, <var>c</var>, <var>n</var>)</em></dt>
<dd><p>Returns the <var>n</var>-th term of the harmonic series
<code><var>a</var>/<var>b</var>, <var>a</var>/(<var>b</var> + <var>c</var>), <var>a</var>/(<var>b</var> + 2*<var>c</var>), ..., <var>a</var>/(<var>b</var> + (<var>n</var> - 1)*<var>c</var>)</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Sums-and-products">Sums and products</a>
·</div>
</dd></dl>
<a name="arithsum"></a><a name="Item_003a-simplifications_002fdeffn_002farithsum"></a><dl>
<dt><a name="index-arithsum"></a>Function: <strong>arithsum</strong> <em>(<var>a</var>, <var>d</var>, <var>n</var>)</em></dt>
<dd><p>Returns the sum of the arithmetic series from 1 to <var>n</var>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Sums-and-products">Sums and products</a>
·</div>
</dd></dl>
<a name="geosum"></a><a name="Item_003a-simplifications_002fdeffn_002fgeosum"></a><dl>
<dt><a name="index-geosum"></a>Function: <strong>geosum</strong> <em>(<var>a</var>, <var>r</var>, <var>n</var>)</em></dt>
<dd><p>Returns the sum of the geometric series from 1 to <var>n</var>. If <var>n</var> is
infinity (<code>inf</code>) then a sum is finite only if the absolute value
of <var>r</var> is less than 1.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Sums-and-products">Sums and products</a>
·</div>
</dd></dl>
<a name="gaussprob"></a><a name="Item_003a-simplifications_002fdeffn_002fgaussprob"></a><dl>
<dt><a name="index-gaussprob"></a>Function: <strong>gaussprob</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the Gaussian probability function
<code>%e^(-<var>x</var>^2/2) / sqrt(2*%pi)</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div>
</dd></dl>
<a name="gd"></a><a name="Item_003a-simplifications_002fdeffn_002fgd"></a><dl>
<dt><a name="index-gd"></a>Function: <strong>gd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the Gudermannian function
<code>2*atan(%e^x)-%pi/2</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div>
</dd></dl>
<a name="agd"></a><a name="Item_003a-simplifications_002fdeffn_002fagd"></a><dl>
<dt><a name="index-agd"></a>Function: <strong>agd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the inverse Gudermannian function
<code>log (tan (%pi/4 + x/2))</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-functs">Package functs</a>
·<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div>
</dd></dl>
<a name="vers"></a><a name="Item_003a-simplifications_002fdeffn_002fvers"></a><dl>
<dt><a name="index-vers"></a>Function: <strong>vers</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the versed sine <code>1 - cos (x)</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
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</dd></dl>
<a name="covers"></a><a name="Item_003a-simplifications_002fdeffn_002fcovers"></a><dl>
<dt><a name="index-covers"></a>Function: <strong>covers</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the coversed sine <code>1 - sin (<var>x</var>)</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
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</dd></dl>
<a name="exsec"></a><a name="Item_003a-simplifications_002fdeffn_002fexsec"></a><dl>
<dt><a name="index-exsec"></a>Function: <strong>exsec</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the exsecant <code>sec (<var>x</var>) - 1</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
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</dd></dl>
<a name="hav"></a><a name="Item_003a-simplifications_002fdeffn_002fhav"></a><dl>
<dt><a name="index-hav"></a>Function: <strong>hav</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Returns the haversine <code>(1 - cos(x))/2</code>.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
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</dd></dl>
<a name="combination"></a><a name="Item_003a-simplifications_002fdeffn_002fcombination"></a><dl>
<dt><a name="index-combination"></a>Function: <strong>combination</strong> <em>(<var>n</var>, <var>r</var>)</em></dt>
<dd><p>Returns the number of combinations of <var>n</var> objects
taken <var>r</var> at a time.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
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</dd></dl>
<a name="permutation"></a><a name="Item_003a-simplifications_002fdeffn_002fpermutation"></a><dl>
<dt><a name="index-permutation"></a>Function: <strong>permutation</strong> <em>(<var>n</var>, <var>r</var>)</em></dt>
<dd><p>Returns the number of permutations of <var>r</var> objects
selected from a set of <var>n</var> objects.
</p>
<p>To use this function write first <code>load("functs")</code>.
</p>
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