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<h1 class="chapter"> 7. Simplification </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC30">7.1 Functions and Variables for Simplification</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
</td></tr>
</table>

<p><a name="Item_003a-Functions-and-Variables-for-Simplification"></a>
</p><hr size="6">
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</tr></table>
<h2 class="section"> 7.1 Functions and Variables for Simplification </h2>

<p><a name="Item_003a-askexp"></a>
</p><dl>
<dt><u>System variable:</u> <b>askexp</b>
<a name="IDX196"></a>
</dt>
<dd><p>When <code>asksign</code> is called,
<code>askexp</code> is the expression <code>asksign</code> is testing.
</p>
<p>At one time, it was possible for a user to inspect <code>askexp</code>
by entering a Maxima break with control-A.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Declarations-and-inferences">Declarations and inferences</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-askinteger"></a>
</p><dl>
<dt><u>Function:</u> <b>askinteger</b><i> (<var>expr</var>, integer)</i>
<a name="IDX197"></a>
</dt>
<dt><u>Function:</u> <b>askinteger</b><i> (<var>expr</var>)</i>
<a name="IDX198"></a>
</dt>
<dt><u>Function:</u> <b>askinteger</b><i> (<var>expr</var>, even)</i>
<a name="IDX199"></a>
</dt>
<dt><u>Function:</u> <b>askinteger</b><i> (<var>expr</var>, odd)</i>
<a name="IDX200"></a>
</dt>
<dd><p><code>askinteger (<var>expr</var>, integer)</code> attempts to determine from the <code>assume</code> database
whether <var>expr</var> is an integer.
<code>askinteger</code> prompts the user if it cannot tell otherwise,
and attempt to install the information in the database if possible.
<code>askinteger (<var>expr</var>)</code> is equivalent to <code>askinteger (<var>expr</var>, integer)</code>.
</p>
<p><code>askinteger (<var>expr</var>, even)</code> and <code>askinteger (<var>expr</var>, odd)</code>
likewise attempt to determine if <var>expr</var> is an even integer or odd integer, respectively.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Declarations-and-inferences">Declarations and inferences</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-asksign"></a>
</p><dl>
<dt><u>Function:</u> <b>asksign</b><i> (<var>expr</var>)</i>
<a name="IDX201"></a>
</dt>
<dd><p>First attempts to determine whether the specified
expression is positive, negative, or zero.  If it cannot, it asks the
user the necessary questions to complete its deduction.  The user's
answer is recorded in the data base for the duration of the current
computation. The return value of <code>asksign</code> is one of <code>pos</code>, <code>neg</code>,
or <code>zero</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Declarations-and-inferences">Declarations and inferences</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-demoivre"></a>
</p><dl>
<dt><u>Function:</u> <b>demoivre</b><i> (<var>expr</var>)</i>
<a name="IDX202"></a>
</dt>
<dt><u>Option variable:</u> <b>demoivre</b>
<a name="IDX203"></a>
</dt>
<dd><p>The function <code>demoivre (expr)</code> converts one expression
without setting the global variable <code>demoivre</code>.
</p>
<p>When the variable <code>demoivre</code> is <code>true</code>,
complex exponentials are converted into equivalent expressions in terms of circular functions:
<code>exp (a + b*%i)</code> simplifies to <code>%e^a * (cos(b) + %i*sin(b))</code>
if <code>b</code> is free of <code>%i</code>.
<code>a</code> and <code>b</code> are not expanded.
</p>
<p>The default value of <code>demoivre</code> is <code>false</code>.
</p>
<p><code>exponentialize</code> converts circular and hyperbolic functions to exponential form.
<code>demoivre</code> and <code>exponentialize</code> cannot
both be true at the same time.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Complex-variables">Complex variables</a>
 &middot;
<a href="maxima_95.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Hyperbolic-functions">Hyperbolic functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-distribute_005fover"></a>
</p><dl>
<dt><u>Option variable:</u> <b>distribute_over</b>
<a name="IDX204"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>distribute_over</code> controls the mapping of functions over bags like lists, 
matrices, and equations. At this time this feature is implemented for the
trigonometric functions, the exponential integrals, and the integer 
functions like <code>mod</code>, <code>floor</code>, <code>ceiling</code>, <code>round</code>.
</p>
<p>The mapping of functions is switched off, when setting <code>distribute_over</code> 
to the value <code>false</code>.
</p>
<p>Examples:
</p>
<p>The <code>sin</code> function maps over a list:
</p>
<pre class="example">(%i1) sin([x,1,1.0]);
(%o1)                 [sin(x), sin(1), .8414709848078965]
</pre>
<p><code>mod</code> is a function with two arguments which maps over lists. Mapping over 
nested lists is possible too:
</p>
<pre class="example">(%i2) mod([x,11,2*a],10);
(%o2)                    [mod(x, 10), 1, 2 mod(a, 5)]
(%i3) mod([[x,y,z],11,2*a],10);
(%o3)       [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]
</pre>
<p>Mapping of the <code>floor</code> function over a matrix and an equation:
</p>
<pre class="example">(%i4) floor(matrix([a,b],[c,d]));
                            [ floor(a)  floor(b) ]
(%o4)                       [                    ]
                            [ floor(c)  floor(d) ]
(%i5) floor(a=b);
(%o5)                         floor(a) = floor(b)
</pre>
<p>Functions with more than one argument map over any of the arguments or all
arguments:
</p>
<pre class="example">(%i6) expintegral_e([1,2],[x,y]);
(%o6) [[expintegral_e(1, x), expintegral_e(1, y)], 
       [expintegral_e(2, x), expintegral_e(2, y)]]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-domain"></a>
</p><dl>
<dt><u>Option variable:</u> <b>domain</b>
<a name="IDX205"></a>
</dt>
<dd><p>Default value: <code>real</code>
</p>
<p>When <code>domain</code> is set to <code>complex</code>, <code>sqrt (x^2)</code> will remain
<code>sqrt (x^2)</code> instead of returning <code>abs(x)</code>.
</p>

<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-expand"></a>
</p><dl>
<dt><u>Function:</u> <b>expand</b><i> (<var>expr</var>)</i>
<a name="IDX206"></a>
</dt>
<dt><u>Function:</u> <b>expand</b><i> (<var>expr</var>, <var>p</var>, <var>n</var>)</i>
<a name="IDX207"></a>
</dt>
<dd><p>Expand expression <var>expr</var>.
Products of sums and exponentiated sums are
multiplied out, numerators of rational expressions which are sums are
split into their respective terms, and multiplication (commutative
and non-commutative) are distributed over addition at all levels of
<var>expr</var>.
</p>
<p>For polynomials one should usually use <code>ratexpand</code> which uses a
more efficient algorithm.
</p>
<p><code>maxnegex</code> and <code>maxposex</code> control the maximum negative and
positive exponents, respectively, which will expand.
</p>
<p><code>expand (<var>expr</var>, <var>p</var>, <var>n</var>)</code> expands <var>expr</var>, 
using <var>p</var> for <code>maxposex</code> and <var>n</var> for <code>maxnegex</code>.
This is useful in order to expand part but not all of an expression.
</p>
<p><code>expon</code> - the exponent of the largest negative power which is
automatically expanded (independent of calls to <code>expand</code>).  For example
if <code>expon</code> is 4 then <code>(x+1)^(-5)</code> will not be automatically expanded.
</p>
<p><code>expop</code> - the highest positive exponent which is automatically
expanded.  Thus <code>(x+1)^3</code>, when typed, will be automatically expanded
only if <code>expop</code> is greater than or equal to 3.  If it is desired to have
<code>(x+1)^n</code> expanded where <code>n</code> is greater than <code>expop</code> then executing
<code>expand ((x+1)^n)</code> will work only if <code>maxposex</code> is not less than <code>n</code>.
</p>
<p><code>expand(expr, 0, 0)</code> causes a resimplification of <code>expr</code>. <code>expr</code>
is not reevaluated. In distinction from <code>ev(expr, noeval)</code> a special 
representation (e. g. a CRE form) is removed. See also <code>ev</code>.
</p>
<p>The <code>expand</code> flag used with <code>ev</code> causes expansion.
</p>
<p>The file <tt>`simplification/facexp.mac'</tt>
contains several related functions (in particular <code>facsum</code>, <code>factorfacsum</code>
and <code>collectterms</code>, which are autoloaded) and variables (<code>nextlayerfactor</code>
and <code>facsum_combine</code>) that provide the user with the ability to structure
expressions by controlled expansion.
Brief function descriptions are available in <tt>`simplification/facexp.usg'</tt>.
A demo is available by doing <code>demo(&quot;facexp&quot;)</code>.
</p>
<p>Examples:
</p><pre class="example">(%i1) expr:(x+1)^2*(y+1)^3;
                                      2        3
(%o1)                          (x + 1)  (y + 1)
(%i2) expand(expr);
       2  3        3    3      2  2        2      2      2
(%o2) x  y  + 2 x y  + y  + 3 x  y  + 6 x y  + 3 y  + 3 x  y
                                                 2
                                + 6 x y + 3 y + x  + 2 x + 1

(%i3) expand(expr,2);
                      2        3              3          3
(%o3)                x  (y + 1)  + 2 x (y + 1)  + (y + 1)

(%i4) expr:(x+1)^-2*(y+1)^3;
                                          3
                                   (y + 1)
(%o4)                              --------
                                          2
                                   (x + 1)
(%i5) expand(expr);
                 3               2
                y             3 y            3 y             1
(%o5)      ------------ + ------------ + ------------ + ------------
            2              2              2              2
           x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1

(%i6) expand(expr,2,2);
                                          3
                                   (y + 1)
(%o6)                            ------------
                                  2
                                 x  + 2 x + 1
</pre>
<p>Resimplify an expression without expansion:
</p>
<pre class="example">(%i7) expr:(1+x)^2*sin(x);
                                       2
(%o7)                           (x + 1)  sin(x)
(%i8) exponentialize:true;
(%o8)                                true
(%i9) expand(expr,0,0);
                                   2    %i x     - %i x
                         %i (x + 1)  (%e     - %e      )
(%o9)                  - -------------------------------
                                        2
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-expandwrt"></a>
</p><dl>
<dt><u>Function:</u> <b>expandwrt</b><i> (<var>expr</var>, <var>x_1</var>, ..., <var>x_n</var>)</i>
<a name="IDX208"></a>
</dt>
<dd><p>Expands expression <code>expr</code> with respect to the 
variables <var>x_1</var>, ..., <var>x_n</var>.
All products involving the variables appear explicitly.  The form returned
will be free of products of sums of expressions that are not free of
the variables.   <var>x_1</var>, ..., <var>x_n</var>
may be variables, operators, or expressions.
</p>
<p>By default, denominators are not expanded, but this can be controlled by
means of the switch <code>expandwrt_denom</code>.
</p>
<p>This function is autoloaded from
<tt>`simplification/stopex.mac'</tt>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-expandwrt_005fdenom"></a>
</p><dl>
<dt><u>Option variable:</u> <b>expandwrt_denom</b>
<a name="IDX209"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>expandwrt_denom</code> controls the treatment of rational
expressions by <code>expandwrt</code>.  If <code>true</code>, then both the numerator and
denominator of the expression will be expanded according to the
arguments of <code>expandwrt</code>, but if <code>expandwrt_denom</code> is <code>false</code>, then only the
numerator will be expanded in that way.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-expandwrt_005ffactored"></a>
</p><dl>
<dt><u>Function:</u> <b>expandwrt_factored</b><i> (<var>expr</var>, <var>x_1</var>, ..., <var>x_n</var>)</i>
<a name="IDX210"></a>
</dt>
<dd><p>is similar to <code>expandwrt</code>, but treats expressions that are products somewhat differently.
<code>expandwrt_factored</code> expands only on those factors of <code>expr</code>
that contain the variables <var>x_1</var>, ..., <var>x_n</var>.
</p>
<p>This function is autoloaded from <tt>`simplification/stopex.mac'</tt>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-expon"></a>
</p><dl>
<dt><u>Option variable:</u> <b>expon</b>
<a name="IDX211"></a>
</dt>
<dd><p>Default value: 0
</p>
<p><code>expon</code> is the exponent of the largest negative power which
is automatically expanded (independent of calls to <code>expand</code>).  For
example, if <code>expon</code> is 4 then <code>(x+1)^(-5)</code> will not be automatically
expanded.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
</p>
</div>


</dd></dl>


<p><a name="Item_003a-exponentialize"></a>
</p><dl>
<dt><u>Function:</u> <b>exponentialize</b><i> (<var>expr</var>)</i>
<a name="IDX212"></a>
</dt>
<dt><u>Option variable:</u> <b>exponentialize</b>
<a name="IDX213"></a>
</dt>
<dd><p>The function <code>exponentialize (expr)</code> converts 
circular and hyperbolic functions in <var>expr</var> to exponentials,
without setting the global variable <code>exponentialize</code>.
</p>
<p>When the variable <code>exponentialize</code> is <code>true</code>,
all circular and hyperbolic functions are converted to exponential form.
The default value is <code>false</code>.
</p>
<p><code>demoivre</code> converts complex exponentials into circular functions.
<code>exponentialize</code> and <code>demoivre</code> cannot
both be true at the same time.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Complex-variables">Complex variables</a>
 &middot;
<a href="maxima_95.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Hyperbolic-functions">Hyperbolic functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-expop"></a>
</p><dl>
<dt><u>Option variable:</u> <b>expop</b>
<a name="IDX214"></a>
</dt>
<dd><p>Default value: 0
</p>
<p><code>expop</code> is the highest positive exponent which is
automatically expanded.  Thus <code>(x + 1)^3</code>, when typed, will be
automatically expanded only if <code>expop</code> is greater than or equal to 3.
If it is desired to have <code>(x + 1)^n</code> expanded where <code>n</code> is greater than
<code>expop</code> then executing <code>expand ((x + 1)^n)</code> will work only if <code>maxposex</code> is
not less than n.
</p>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-factlim"></a>
</p><dl>
<dt><u>Option variable:</u> <b>factlim</b>
<a name="IDX215"></a>
</dt>
<dd><p>Default value: -1
</p>
<p><code>factlim</code> specifies the highest factorial which is
automatically expanded.  If it is -1 then all integers are expanded.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-intosum"></a>
</p><dl>
<dt><u>Function:</u> <b>intosum</b><i> (<var>expr</var>)</i>
<a name="IDX216"></a>
</dt>
<dd><p>Moves multiplicative factors outside a summation to inside.
If the index is used in the
outside expression, then the function tries to find a reasonable
index, the same as it does for <code>sumcontract</code>.  This is essentially the
reverse idea of the <code>outative</code> property of summations, but note that it
does not remove this property, it only bypasses it.
</p>
<p>In some cases,
a <code>scanmap (multthru, <var>expr</var>)</code> may be necessary before the <code>intosum</code>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-lassociative"></a>
</p><dl>
<dt><u>Declaration:</u> <b>lassociative</b>
<a name="IDX217"></a>
</dt>
<dd><p><code>declare (g, lassociative)</code> tells the
Maxima simplifier that <code>g</code> is left-associative.  E.g., <code>g (g (a, b), g (c, d))</code> will
simplify to <code>g (g (g (a, b), c), d)</code>.
</p>
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 &middot;
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 &middot;
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-linear"></a>
</p><dl>
<dt><u>Declaration:</u> <b>linear</b>
<a name="IDX218"></a>
</dt>
<dd><p>One of Maxima's operator properties.  For univariate <code>f</code> so
declared, &quot;expansion&quot; <code>f(x + y)</code> yields <code>f(x) + f(y)</code>,
<code>f(a*x)</code> yields <code>a*f(x)</code> takes
place where <code>a</code> is a &quot;constant&quot;.  For functions of two or more arguments,
&quot;linearity&quot; is defined to be as in the case of <code>sum</code> or <code>integrate</code>,
i.e., <code>f (a*x + b, x)</code> yields <code>a*f(x,x) + b*f(1,x)</code>
for <code>a</code> and <code>b</code> free of <code>x</code>.
</p>
<p><code>linear</code> is equivalent to <code>additive</code> and <code>outative</code>.
See also <code>opproperties</code>.
</p>
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 &middot;
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 &middot;
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-mainvar"></a>
</p><dl>
<dt><u>Declaration:</u> <b>mainvar</b>
<a name="IDX219"></a>
</dt>
<dd><p>You may declare variables to be <code>mainvar</code>.  The ordering
scale for atoms is essentially: numbers &lt; constants (e.g., <code>%e</code>, <code>%pi</code>) &lt;
scalars &lt; other variables &lt; mainvars.  E.g., compare <code>expand ((X+Y)^4)</code>
with <code>(declare (x, mainvar), expand ((x+y)^4))</code>.  (Note: Care should be
taken if you elect to use the above feature.  E.g., if you subtract an
expression in which <code>x</code> is a <code>mainvar</code> from one in which <code>x</code> isn't a
<code>mainvar</code>, resimplification e.g. with <code>ev (expr, simp)</code> may be
necessary if cancellation is to occur.  Also, if you save an
expression in which <code>x</code> is a <code>mainvar</code>, you probably should also save <code>x</code>.)
</p>
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 &middot;
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-maxapplydepth"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxapplydepth</b>
<a name="IDX220"></a>
</dt>
<dd><p>Default value: 10000
</p>
<p><code>maxapplydepth</code> is the maximum depth to which <code>apply1</code>
and <code>apply2</code> will delve.
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-maxapplyheight"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxapplyheight</b>
<a name="IDX221"></a>
</dt>
<dd><p>Default value: 10000
</p>
<p><code>maxapplyheight</code> is the maximum height to which <code>applyb1</code>
will reach before giving up.
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-maxnegex"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxnegex</b>
<a name="IDX222"></a>
</dt>
<dd><p>Default value: 1000
</p>
<p><code>maxnegex</code> is the largest negative exponent which will
be expanded by the <code>expand</code> command (see also <code>maxposex</code>).
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-maxposex"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxposex</b>
<a name="IDX223"></a>
</dt>
<dd><p>Default value: 1000
</p>
<p><code>maxposex</code> is the largest exponent which will be
expanded with the <code>expand</code> command (see also <code>maxnegex</code>).
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-multiplicative"></a>
</p><dl>
<dt><u>Declaration:</u> <b>multiplicative</b>
<a name="IDX224"></a>
</dt>
<dd><p><code>declare (f, multiplicative)</code> tells the Maxima simplifier that <code>f</code> is multiplicative.
</p>
<ol>
<li>
If <code>f</code> is univariate, whenever the simplifier encounters <code>f</code> applied
to a product, <code>f</code> distributes over that product.  E.g., <code>f(x*y)</code>
simplifies to <code>f(x)*f(y)</code>.
</li><li>
If <code>f</code> is a function of 2 or more arguments, multiplicativity is
defined as multiplicativity in the first argument to <code>f</code>, e.g.,
<code>f (g(x) * h(x), x)</code> simplifies to <code>f (g(x) ,x) * f (h(x), x)</code>.
</li></ol>

<p>This simplification does not occur when <code>f</code> is applied to expressions of
the form <code>product (x[i], i, m, n)</code>.
</p>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Declarations-and-inferences">Declarations and inferences</a>
 &middot;
<a href="maxima_95.html#Category_003a-Expressions">Expressions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification">Simplification</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-negdistrib"></a>
</p><dl>
<dt><u>Option variable:</u> <b>negdistrib</b>
<a name="IDX225"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>negdistrib</code> is <code>true</code>, -1 distributes
over an expression.  E.g., <code>-(x + y)</code> becomes <code>- y - x</code>.  Setting it to <code>false</code>
will allow <code>- (x + y)</code> to be displayed like that.  This is sometimes useful
but be very careful: like the <code>simp</code> flag, this is one flag you do not
want to set to <code>false</code> as a matter of course or necessarily for other
than local use in your Maxima.
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-negsumdispflag"></a>
</p><dl>
<dt><u>Option variable:</u> <b>negsumdispflag</b>
<a name="IDX226"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>negsumdispflag</code> is <code>true</code>, <code>x - y</code> displays as <code>x - y</code>
instead of as <code>- y + x</code>.  Setting it to <code>false</code> causes the special check in
display for the difference of two expressions to not be done.  One
application is that thus <code>a + %i*b</code> and <code>a - %i*b</code> may both be displayed the
same way.
</p>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Display-flags-and-variables">Display flags and variables</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-noeval"></a>
</p><dl>
<dt><u>Special symbol:</u> <b>noeval</b>
<a name="IDX227"></a>
</dt>
<dd><p><code>noeval</code> suppresses the evaluation phase of <code>ev</code>.  This is useful in
conjunction with other switches and in causing expressions      
to be resimplified without being reevaluated.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-noun"></a>
</p><dl>
<dt><u>Declaration:</u> <b>noun</b>
<a name="IDX228"></a>
</dt>
<dd><p><code>noun</code> is one of the options of the <code>declare</code> command.  It makes a
function so declared a &quot;noun&quot;, meaning that it won't be evaluated
automatically.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Nouns-and-verbs">Nouns and verbs</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-noundisp"></a>
</p><dl>
<dt><u>Option variable:</u> <b>noundisp</b>
<a name="IDX229"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>noundisp</code> is <code>true</code>, nouns display with
a single quote.  This switch is always <code>true</code> when displaying function
definitions.
</p>
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 &middot;
<a href="maxima_95.html#Category_003a-Nouns-and-verbs">Nouns and verbs</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-nouns"></a>
</p><dl>
<dt><u>Special symbol:</u> <b>nouns</b>
<a name="IDX230"></a>
</dt>
<dd><p><code>nouns</code> is an <code>evflag</code>. When used as an option to the <code>ev</code> command,
<code>nouns</code> converts all
&quot;noun&quot; forms occurring in the expression being <code>ev</code>'d to &quot;verbs&quot;, i.e.,
evaluates them.  See also <code>noun</code>, <code>nounify</code>, <code>verb</code>, and <code>verbify</code>.
</p>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Evaluation-flags">Evaluation flags</a>
 &middot;
<a href="maxima_95.html#Category_003a-Nouns-and-verbs">Nouns and verbs</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-numer"></a>
</p><dl>
<dt><u>Option variable:</u> <b>numer</b>
<a name="IDX231"></a>
</dt>
<dd><p><code>numer</code> causes some mathematical functions (including exponentiation)
with numerical arguments to be evaluated in floating point. It causes
variables in <code>expr</code> which have been given numerals to be replaced by
their values.  It also sets the <code>float</code> switch on.
</p>
<p>See also <code>%enumer</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
                                               1
(%o1)                   [sqrt(2), sin(1), -----------]
                                          sqrt(3) + 1
(%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer;
(%o2)      [1.414213562373095, .8414709848078965, .3660254037844387]
</pre>
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 &middot;
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</p>
</div>

</dd></dl>


<p><a name="Item_003a-numerval"></a>
</p><dl>
<dt><u>Function:</u> <b>numerval</b><i> (<var>x_1</var>, <var>expr_1</var>, ..., <var>var_n</var>, <var>expr_n</var>)</i>
<a name="IDX232"></a>
</dt>
<dd><p>Declares the variables <code>x_1</code>, ..., <var>x_n</var> to have
numeric values equal to <code>expr_1</code>, ..., <code>expr_n</code>.
The numeric value is evaluated and substituted for the variable
in any expressions in which the variable occurs if the <code>numer</code> flag is
<code>true</code>. See also <code>ev</code>.
</p>
<p>The expressions <code>expr_1</code>, ..., <code>expr_n</code> can be any expressions,
not necessarily numeric.
</p>
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</p>
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</dd></dl>


<p><a name="Item_003a-opproperties"></a>
</p><dl>
<dt><u>System variable:</u> <b>opproperties</b>
<a name="IDX233"></a>
</dt>
<dd><p><code>opproperties</code> is the list of the special operator properties recognized by
the Maxima simplifier:
<code>linear</code>, <code>additive</code>, <code>multiplicative</code>, <code>outative</code>, <code>evenfun</code>,
<code>oddfun</code>, <code>commutative</code>, <code>symmetric</code>, <code>antisymmetric</code>, <code>nary</code>, 
<code>lassociative</code>, <code>rassociative</code>.
</p>
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 &middot;
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</p>
</div>


</dd></dl>


<p><a name="Item_003a-opsubst"></a>
</p><dl>
<dt><u>Option variable:</u> <b>opsubst</b>
<a name="IDX234"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>opsubst</code> is <code>false</code>, <code>subst</code> does not attempt to
substitute into the operator of an expression.  E.g., 
<code>(opsubst: false, subst (x^2, r, r+r[0]))</code> will work.
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-outative"></a>
</p><dl>
<dt><u>Declaration:</u> <b>outative</b>
<a name="IDX235"></a>
</dt>
<dd><p><code>declare (f, outative)</code> tells the Maxima simplifier that constant factors
in the argument of <code>f</code> can be pulled out.
</p>
<ol>
<li>
If <code>f</code> is univariate, whenever the simplifier encounters <code>f</code> applied
to a product, that product will be partitioned into factors that are
constant and factors that are not and the constant factors will be
pulled out.  E.g., <code>f(a*x)</code> will simplify to <code>a*f(x)</code> where <code>a</code> is a
constant.  Non-atomic constant factors will not be pulled out.
</li><li>
If <code>f</code> is a function of 2 or more arguments, outativity is defined
as in the case of <code>sum</code> or <code>integrate</code>, i.e., <code>f (a*g(x), x)</code> will simplify
to <code>a * f(g(x), x)</code> for <code>a</code> free of <code>x</code>.
</li></ol>

<p><code>sum</code>, <code>integrate</code>, and <code>limit</code> are all <code>outative</code>.
</p>
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 &middot;
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-posfun"></a>
</p><dl>
<dt><u>Declaration:</u> <b>posfun</b>
<a name="IDX236"></a>
</dt>
<dd><p><code>declare (f, posfun)</code> declares <code>f</code> to be a positive function.
<code>is (f(x) &gt; 0)</code> yields <code>true</code>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-pred"></a>
</p><dl>
<dt><u>Special symbol:</u> <b>pred</b>
<a name="IDX237"></a>
</dt>
<dd><p>As an argument in a call to <code>ev (<var>expr</var>)</code>, <code>pred</code> causes 
predicates (expressions which evaluate to <code>true</code> or <code>false</code>) to be 
evaluated. See <code>ev</code>.
</p>
<p>Example:
</p>
<pre class="example">(%i1) 1&lt;2;
(%o1)                                1 &lt; 2
(%i2) 1&lt;2,pred;
(%o2)                                true
</pre>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-radcan"></a>
</p><dl>
<dt><u>Function:</u> <b>radcan</b><i> (<var>expr</var>)</i>
<a name="IDX238"></a>
</dt>
<dd><p>Simplifies <var>expr</var>, which can contain logs, exponentials, and radicals, by 
converting it into a form which is canonical over a large class of expressions 
and a given ordering of variables; that is, all functionally equivalent forms 
are mapped into a unique form.  For a somewhat larger class of expressions, 
<code>radcan</code> produces a regular form. Two equivalent expressions in this class 
do not necessarily have the same appearance, but their difference can be 
simplified by <code>radcan</code> to zero.
</p>
<p>For some expressions <code>radcan</code> is quite time consuming.  This is the cost 
of exploring certain relationships among the components of the expression for 
simplifications based on factoring and partial-fraction expansions of exponents.  
</p>



<p>Examples:
</p>
<pre class="example">(%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
                                           a/2
(%o1)                            log(x + 1)

(%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
(%o2)                                  2

(%i3) radcan((%e^x-1)/(1+%e^(x/2)));
                                     x/2
(%o3)                              %e    - 1
</pre>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-radexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>radexpand</b>
<a name="IDX239"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>radexpand</code> controls some simplifications of radicals.
</p>
<p>When <code>radexpand</code> is <code>all</code>, causes nth roots of
factors of a product which are powers of n to be pulled outside of the
radical.  E.g. if <code>radexpand</code> is <code>all</code>, <code>sqrt (16*x^2)</code> simplifies to <code>4*x</code>.
</p>
<p>More particularly, consider <code>sqrt (x^2)</code>.
</p><ul>
<li>
If <code>radexpand</code> is <code>all</code> or <code>assume (x &gt; 0)</code> has been executed, 
<code>sqrt(x^2)</code> simplifies to <code>x</code>.
</li><li>
If <code>radexpand</code> is <code>true</code> and <code>domain</code> is <code>real</code> (its default), 
<code>sqrt(x^2)</code> simplifies to <code>abs(x)</code>.
</li><li>
If <code>radexpand</code> is <code>false</code>, or <code>radexpand</code> is <code>true</code> and <code>domain</code> is <code>complex</code>, 
<code>sqrt(x^2)</code> is not simplified.
</li></ul>

<p>Note that <code>domain</code> only matters when <code>radexpand</code> is <code>true</code>.
</p>
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</div>


</dd></dl>


<p><a name="Item_003a-radsubstflag"></a>
</p><dl>
<dt><u>Option variable:</u> <b>radsubstflag</b>
<a name="IDX240"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>radsubstflag</code>, if <code>true</code>, permits <code>ratsubst</code> to make
substitutions such as <code>u</code> for <code>sqrt (x)</code> in <code>x</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-rassociative"></a>
</p><dl>
<dt><u>Declaration:</u> <b>rassociative</b>
<a name="IDX241"></a>
</dt>
<dd><p><code>declare (g, rassociative)</code> tells the Maxima
simplifier that <code>g</code> is right-associative.  E.g.,
<code>g(g(a, b), g(c, d))</code> simplifies to <code>g(a, g(b, g(c, d)))</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Declarations-and-inferences">Declarations and inferences</a>
 &middot;
<a href="maxima_95.html#Category_003a-Operators">Operators</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-scsimp"></a>
</p><dl>
<dt><u>Function:</u> <b>scsimp</b><i> (<var>expr</var>, <var>rule_1</var>, ..., <var>rule_n</var>)</i>
<a name="IDX242"></a>
</dt>
<dd><p>Sequential Comparative Simplification (method due to Stoute).
<code>scsimp</code> attempts to simplify <var>expr</var>
according to the rules <var>rule_1</var>, ..., <var>rule_n</var>.
If a smaller expression is obtained, the process
repeats.  Otherwise after all simplifications are tried, it returns
the original answer.
</p>
<p><code>example (scsimp)</code> displays some examples.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Simplification-functions">Simplification functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-simp"></a>
</p><dl>
<dt><u>Option variable:</u> <b>simp</b>
<a name="IDX243"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>simp</code> enables simplification. This is the standard. <code>simp</code> is also 
an <code>evflag</code>, which is recognized by the function <code>ev</code>. See <code>ev</code>.
</p>
<p>When <code>simp</code> is used as an <code>evflag</code> with a value <code>false</code>, the 
simplification is suppressed only during the evaluation phase of an expression. 
The flag can not suppress the simplification which follows the evaluation 
phase.
</p>
<p>Examples:
</p>
<p>The simplification is switched off globally. The expression <code>sin(1.0)</code> is 
not simplified to its numerical value. The <code>simp</code>-flag switches the 
simplification on.
</p>
<pre class="example">(%i1) simp:false;
(%o1)                                false
(%i2) sin(1.0);
(%o2)                              sin(1.0)
(%i3) sin(1.0),simp;
(%o3)                          .8414709848078965
</pre>
<p>The simplification is switched on again. The <code>simp</code>-flag cannot suppress 
the simplification completely. The output shows a simplified expression, but 
the variable <code>x</code> has an unsimplified expression as a value, because the 
assignment has occurred during the evaluation phase of the expression.
</p>
<pre class="example">(%i4) simp:true;
(%o4)                                true
(%i5) x:sin(1.0),simp:false;
(%o5)                          .8414709848078965
(%i6) :lisp $X
((%SIN) 1.0)
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Evaluation-flags">Evaluation flags</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-simpsum"></a>
</p><dl>
<dt><u>Option variable:</u> <b>simpsum</b>
<a name="IDX244"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>simpsum</code> is <code>true</code>, the result of a <code>sum</code> is
simplified.  This simplification may sometimes be able to produce a
closed form.  If <code>simpsum</code> is <code>false</code> or if the quoted form <code>'sum</code> is used, the value is a
sum noun form which is a representation of the sigma notation used in
mathematics.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-sumcontract"></a>
</p><dl>
<dt><u>Function:</u> <b>sumcontract</b><i> (<var>expr</var>)</i>
<a name="IDX245"></a>
</dt>
<dd><p>Combines all sums of an addition that have
upper and lower bounds that differ by constants. The result is an
expression containing one summation for each set of such summations
added to all appropriate extra terms that had to be extracted to form
this sum.  <code>sumcontract</code> combines all compatible sums and uses one of
the indices from one of the sums if it can, and then try to form a
reasonable index if it cannot use any supplied.
</p>
<p>It may be necessary to do an <code>intosum (<var>expr</var>)</code> before the <code>sumcontract</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-sumexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>sumexpand</b>
<a name="IDX246"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>sumexpand</code> is <code>true</code>, products of sums and
exponentiated sums simplify to nested sums.
</p>
<p>See also <code>cauchysum</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) sumexpand: true$
(%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                     m      n
                    ====   ====
                    \      \
(%o2)                &gt;      &gt;     f(i1) g(i2)
                    /      /
                    ====   ====
                    i1 = 0 i2 = 0
(%i3) sum (f (i), i, 0, m)^2;
                     m      m
                    ====   ====
                    \      \
(%o3)                &gt;      &gt;     f(i3) f(i4)
                    /      /
                    ====   ====
                    i3 = 0 i4 = 0
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-sumsplitfact"></a>
</p><dl>
<dt><u>Option variable:</u> <b>sumsplitfact</b>
<a name="IDX247"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>sumsplitfact</code> is <code>false</code>,
<code>minfactorial</code> is applied after a <code>factcomb</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-symmetric"></a>
</p><dl>
<dt><u>Declaration:</u> <b>symmetric</b>
<a name="IDX248"></a>
</dt>
<dd><p><code>declare (h, symmetric)</code> tells the Maxima
simplifier that <code>h</code> is a symmetric function.  E.g., <code>h (x, z, y)</code> 
simplifies to <code>h (x, y, z)</code>.
</p>
<p><code>commutative</code> is synonymous with <code>symmetric</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Declarations-and-inferences">Declarations and inferences</a>
 &middot;
<a href="maxima_95.html#Category_003a-Operators">Operators</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-unknown"></a>
</p><dl>
<dt><u>Function:</u> <b>unknown</b><i> (<var>expr</var>)</i>
<a name="IDX249"></a>
</dt>
<dd><p>Returns <code>true</code> if and only if <var>expr</var> contains an operator or function
not recognized by the Maxima simplifier.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification-functions">Simplification functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-Plotting"></a>
</p><hr size="6">
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