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<h1 class="chapter"> 6. Expressions </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC22">6.1 Introduction to Expressions</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC23">6.2 Assignment</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                  
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC24">6.3 Complex</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                     
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC25">6.4 Nouns and Verbs</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC26">6.5 Identifiers</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC27">6.6 Strings</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC28">6.7 Inequality</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                  
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC29">6.8 Syntax</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                      
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC30">6.9 Definitions for Expressions</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
</td></tr>
</table>

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<h2 class="section"> 6.1 Introduction to Expressions </h2>

<p>There are a number of reserved words which cannot be used as
variable names.   Their use would cause a possibly cryptic syntax error.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">integrate            next           from                 diff            
in                   at             limit                sum             
for                  and            elseif               then            
else                 do             or                   if              
unless               product        while                thru            
step                                                                     
</pre></td></tr></table>
<p>Most things in Maxima are expressions.   A sequence of expressions
can be made into an expression by separating them by commas and
putting parentheses around them.   This is similar to the <b>C</b>
<i>comma expression</i>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) x: 3$
(%i2) (x: x+1, x: x^2);
(%o2)                          16
(%i3) (if (x &gt; 17) then 2 else 4);
(%o3)                           4
(%i4) (if (x &gt; 17) then x: 2 else y: 4, y+x);
(%o4)                          20
</pre></td></tr></table>
<p>Even loops in Maxima are expressions, although the value they
return is the not too useful <code>done</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$
(%i2) y;
(%o2)                         done
</pre></td></tr></table>
<p>whereas what you really want is probably to include a third
term in the <i>comma expression</i> which actually gives back the value.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$
(%i4) y;
(%o4)                        3628800
</pre></td></tr></table>


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<h2 class="section"> 6.2 Assignment </h2>
<p>There are two assignment operators in Maxima, <code>:</code> and <code>::</code>.
E.g., <code>a: 3</code> sets the variable <code>a</code> to 3. <code>::</code> assigns the value of the
expression on its right to the value of the quantity on its left,
which must evaluate to an atomic variable or subscripted variable.
</p>
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<h2 class="section"> 6.3 Complex </h2>
<p>A complex expression is specified in Maxima by adding the
real part of the expression to <code>%i</code> times the imaginary part.  Thus the
roots of the equation <code>x^2 - 4*x + 13 = 0</code> are <code>2 + 3*%i</code> and <code>2 - 3*%i</code>.  Note that
simplification of products of complex expressions can be effected by
expanding the product.  Simplification of quotients, roots, and other
functions of complex expressions can usually be accomplished by using
the <code>realpart</code>, <code>imagpart</code>, <code>rectform</code>, <code>polarform</code>, <code>abs</code>, <code>carg</code> functions.
</p>
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<h2 class="section"> 6.4 Nouns and Verbs </h2>
<p>Maxima distinguishes between operators which are &quot;nouns&quot; and operators which are &quot;verbs&quot;.
A verb is an operator which can be executed.
A noun is an operator which appears as a symbol in an expression, without being executed.
By default, function names are verbs.
A verb can be changed into a noun by quoting the function name
or applying the <code>nounify</code> function.
A noun can be changed into a verb by applying the <code>verbify</code> function.
The evaluation flag <code>nouns</code> causes <code>ev</code> to evaluate nouns in an expression.
</p>
<p>The verb form is distinguished by 
a leading dollar sign <code>$</code> on the corresponding Lisp symbol.
In contrast,
the noun form is distinguished by 
a leading percent sign <code>%</code> on the corresponding Lisp symbol.
Some nouns have special display properties, such as <code>'integrate</code> and <code>'derivative</code>
(returned by <code>diff</code>), but most do not.
By default, the noun and verb forms of a function are identical when displayed.
The global flag <code>noundisp</code> causes Maxima to display nouns with a leading quote mark <code>'</code>.
</p>
<p>See also <code>noun</code>, <code>nouns</code>, <code>nounify</code>, and <code>verbify</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) foo (x) := x^2;
                                     2
(%o1)                     foo(x) := x
(%i2) foo (42);
(%o2)                         1764
(%i3) 'foo (42);
(%o3)                        foo(42)
(%i4) 'foo (42), nouns;
(%o4)                         1764
(%i5) declare (bar, noun);
(%o5)                         done
(%i6) bar (x) := x/17;
                                     x
(%o6)                    ''bar(x) := --
                                     17
(%i7) bar (52);
(%o7)                        bar(52)
(%i8) bar (52), nouns;
                               52
(%o8)                          --
                               17
(%i9) integrate (1/x, x, 1, 42);
(%o9)                        log(42)
(%i10) 'integrate (1/x, x, 1, 42);
                             42
                            /
                            [   1
(%o10)                      I   - dx
                            ]   x
                            /
                             1
(%i11) ev (%, nouns);
(%o11)                       log(42)
</pre></td></tr></table>
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<h2 class="section"> 6.5 Identifiers </h2>
<p>Maxima identifiers may comprise alphabetic characters,
plus the numerals 0 through 9,
plus any special character preceded by the backslash <code>\</code> character.
</p>
<p>A numeral may be the first character of an identifier
if it is preceded by a backslash.
Numerals which are the second or later characters need not be preceded by a backslash.
</p>
<p>A special character may be declared alphabetic by the <code>declare</code> function.
If so declared, it need not be preceded by a backslash in an identifier.
The alphabetic characters are initially 
<code>A</code> through <code>Z</code>, <code>a</code> through <code>z</code>, <code>%</code>, and <code>_</code>.
</p>
<p>Maxima is case-sensitive. The identifiers <code>foo</code>, <code>FOO</code>, and <code>Foo</code> are distinct.
See <a href="maxima_3.html#SEC7">Lisp and Maxima</a> for more on this point.
</p>
<p>A Maxima identifier is a Lisp symbol which begins with a dollar sign <code>$</code>.
Any other Lisp symbol is preceded by a question mark <code>?</code> when it appears in Maxima.
See <a href="maxima_3.html#SEC7">Lisp and Maxima</a> for more on this point.
</p>
<p>Examples:
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) %an_ordinary_identifier42;
(%o1)               %an_ordinary_identifier42
(%i2) embedded\ spaces\ in\ an\ identifier;
(%o2)           embedded spaces in an identifier
(%i3) symbolp (%);
(%o3)                         true
(%i4) [foo+bar, foo\+bar];
(%o4)                 [foo + bar, foo+bar]
(%i5) [1729, \1729];
(%o5)                     [1729, 1729]
(%i6) [symbolp (foo\+bar), symbolp (\1729)];
(%o6)                     [true, true]
(%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)];
(%o7)                    [false, false]
(%i8) baz\~quux;
(%o8)                       baz~quux
(%i9) declare (&quot;~&quot;, alphabetic);
(%o9)                         done
(%i10) baz~quux;
(%o10)                      baz~quux
(%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)];
(%o11)                [false, false, false]
(%i12) :lisp (defvar *my-lisp-variable* '$foo)
*MY-LISP-VARIABLE*
(%i12) ?\*my\-lisp\-variable\*;
(%o12)                         foo
</pre></td></tr></table>
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<h2 class="section"> 6.6 Strings </h2>

<p>Strings (quoted character sequences) are enclosed in double quote marks <code>&quot;</code> for input,
and displayed with or without the quote marks, 
depending on the global variable <code>?stringdisp</code>.
</p>
<p>Strings may contain any characters,
including embedded tab, newline, and carriage return characters.
The sequence <code>\&quot;</code> is recognized as a literal double quote,
and <code>\\</code> as a literal backslash.
When backslash appears at the end of a line,
the backslash and the line termination
(either newline or carriage return and newline)
are ignored,
so that the string continues with the next line.
No other special combinations of backslash with another character are recognized;
when backslash appears before any character other than <code>&quot;</code>, <code>\</code>,
or a line termination, the backslash is ignored.
There is no way to represent a special character
(such as tab, newline, or carriage return)
except by embedding the literal character in the string.
</p>
<p>There is no character type in Maxima;
a single character is represented as a one-character string.
</p>
<p>Maxima strings are implemented as Lisp symbols, not Lisp strings;
that may change in a future version of Maxima.
Maxima can display Lisp strings and Lisp characters,
although some other operations (for example, equality tests) may fail.
</p>
<p>The <code>stringproc</code> add-on package contains many functions for working with strings.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) s_1 : &quot;This is a Maxima string.&quot;;
(%o1)               This is a Maxima string.
(%i2) s_2 : &quot;Embedded \&quot;double quotes\&quot; and backslash \\ characters.&quot;;
(%o2) Embedded &quot;double quotes&quot; and backslash \ characters.
(%i3) s_3 : &quot;Embedded line termination
in this string.&quot;;
(%o3) Embedded line termination
in this string.
(%i4) s_4 : &quot;Ignore the \
line termination \
characters in \
this string.&quot;;
(%o4) Ignore the line termination characters in this string.
(%i5) ?stringdisp : false;
(%o5)                         false
(%i6) s_1;
(%o6)               This is a Maxima string.
(%i7) ?stringdisp : true;
(%o7)                         true
(%i8) s_1;
(%o8)              &quot;This is a Maxima string.&quot;
</pre></td></tr></table>
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<h2 class="section"> 6.7 Inequality </h2>
<p>Maxima has the inequality operators <code>&lt;</code>, <code>&lt;=</code>, <code>&gt;=</code>, <code>&gt;</code>, <code>#</code>, and <code>notequal</code>.
See <code>if</code> for a description of conditional expressions.
</p>
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<h2 class="section"> 6.8 Syntax </h2>
<p>It is possible to define new operators with specified precedence,
to undefine existing operators,
or to redefine the precedence of existing operators.  
An operator may be unary prefix or unary postfix, binary infix, n-ary infix, matchfix, or nofix.
&quot;Matchfix&quot; means a pair of symbols which enclose their argument or arguments,
and &quot;nofix&quot; means an operator which takes no arguments.
As examples of the different types of operators, there are the following.
</p>
<dl compact="compact">
<dt> unary prefix</dt>
<dd><p>negation <code>- a</code>
</p></dd>
<dt> unary postfix</dt>
<dd><p>factorial <code>a!</code>
</p></dd>
<dt> binary infix</dt>
<dd><p>exponentiation <code>a^b</code>
</p></dd>
<dt> n-ary infix</dt>
<dd><p>addition <code>a + b</code>
</p></dd>
<dt> matchfix</dt>
<dd><p>list construction <code>[a, b]</code>
</p></dd>
</dl>

<p>(There are no built-in nofix operators;
for an example of such an operator, see <code>nofix</code>.)
</p>
<p>The mechanism to define a new operator is straightforward.
It is only necessary to declare a function as an operator;
the operator function might or might not be defined.
</p>
<p>An example of user-defined operators is the following.
Note that the explicit function call <code>&quot;dd&quot; (a)</code> is equivalent to <code>dd a</code>,
likewise <code>&quot;&lt;-&quot; (a, b)</code> is equivalent to <code>a &lt;- b</code>.
Note also that the functions <code>&quot;dd&quot;</code> and <code>&quot;&lt;-&quot;</code> are undefined in this example.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) prefix (&quot;dd&quot;);
(%o1)                          dd
(%i2) dd a;
(%o2)                         dd a
(%i3) &quot;dd&quot; (a);
(%o3)                         dd a
(%i4) infix (&quot;&lt;-&quot;);
(%o4)                          &lt;-
(%i5) a &lt;- dd b;
(%o5)                      a &lt;- dd b
(%i6) &quot;&lt;-&quot; (a, &quot;dd&quot; (b));
(%o6)                      a &lt;- dd b
</pre></td></tr></table>
<p>The Maxima functions which define new operators are summarized in this table,
stating the default left and right binding powers (lbp and rbp, respectively).
(Binding power determines operator precedence. However, since left and right
binding powers can differ, binding power is somewhat more complicated than precedence.)
Some of the operation definition functions take additional arguments;
see the function descriptions for details.
</p>
<dl compact="compact">
<dt> <code>prefix</code></dt>
<dd><p>rbp=180
</p></dd>
<dt> <code>postfix</code></dt>
<dd><p>lbp=180
</p></dd>
<dt> <code>infix</code></dt>
<dd><p>lbp=180, rbp=180
</p></dd>
<dt> <code>nary</code></dt>
<dd><p>lbp=180, rbp=180
</p></dd>
<dt> <code>matchfix</code></dt>
<dd><p>(binding power not applicable)
</p></dd>
<dt> <code>nofix</code></dt>
<dd><p>(binding power not applicable)
</p></dd>
</dl>

<p>For comparison,
here are some built-in operators and their left and right binding powers.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">Operator   lbp     rbp

  :        180     20 
  ::       180     20 
  :=       180     20 
  ::=      180     20 
  !        160
  !!       160
  ^        140     139 
  .        130     129 
  *        120
  /        120     120 
  +        100     100 
  -        100     134 
  =        80      80 
  #        80      80 
  &gt;        80      80 
  &gt;=       80      80 
  &lt;        80      80 
  &lt;=       80      80 
  not              70 
  and      65
  or       60
  ,        10
  $        -1
  ;        -1
</pre></td></tr></table>
<p><code>remove</code> and <code>kill</code> remove operator properties from an atom.
<code>remove (&quot;<var>a</var>&quot;, op)</code> removes only the operator properties of <var>a</var>.
<code>kill (&quot;<var>a</var>&quot;)</code> removes all properties of <var>a</var>, including the operator properties.
Note that the name of the operator must be enclosed in quotation marks.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) infix (&quot;@&quot;);
(%o1)                           @
(%i2) &quot;@&quot; (a, b) := a^b;
                                     b
(%o2)                      a @ b := a
(%i3) 5 @ 3;
(%o3)                          125
(%i4) remove (&quot;@&quot;, op);
(%o4)                         done
(%i5) 5 @ 3;
Incorrect syntax: @ is not an infix operator
5 @
 ^
(%i5) &quot;@&quot; (5, 3);
(%o5)                          125
(%i6) infix (&quot;@&quot;);
(%o6)                           @
(%i7) 5 @ 3;
(%o7)                          125
(%i8) kill (&quot;@&quot;);
(%o8)                         done
(%i9) 5 @ 3;
Incorrect syntax: @ is not an infix operator
5 @
 ^
(%i9) &quot;@&quot; (5, 3);
(%o9)                        @(5, 3)
</pre></td></tr></table>
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<h2 class="section"> 6.9 Definitions for Expressions </h2>

<dl>
<dt><u>Function:</u> <b>at</b><i> (<var>expr</var>, [<var>eqn_1</var>, ..., <var>eqn_n</var>])</i>
<a name="IDX121"></a>
</dt>
<dt><u>Function:</u> <b>at</b><i> (<var>expr</var>, <var>eqn</var>)</i>
<a name="IDX122"></a>
</dt>
<dd><p>Evaluates the expression <var>expr</var> with
the variables assuming the values as specified for them in the list of
equations <code>[<var>eqn_1</var>, ..., <var>eqn_n</var>]</code> or the single equation <var>eqn</var>.
</p>
<p>If a subexpression depends on any of the variables for which a value is specified
but there is no atvalue specified and it can't be otherwise evaluated,
then a noun form of the <code>at</code> is returned which displays in a two-dimensional form.
</p>
<p><code>at</code> carries out multiple substitutions in series, not parallel.
</p>
<p>See also <code>atvalue</code>.
For other functions which carry out substitutions,
see also <code>subst</code> and <code>ev</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                2
(%o1)                          a
(%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
(%o2)                        @2 + 1
(%i3) printprops (all, atvalue);
                                !
                  d             !
                 --- (f(@1, @2))!       = @2 + 1
                 d@1            !
                                !@1 = 0

                                     2
                          f(0, 1) = a

(%o3)                         done
(%i4) diff (4*f(x, y)^2 - u(x, y)^2, x);
                  d                          d
(%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                  dx                         dx
(%i5) at (%, [x = 0, y = 1]);
                                         !
              2              d           !
(%o5)     16 a  - 2 u(0, 1) (-- (u(x, y))!            )
                             dx          !
                                         !x = 0, y = 1
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>box</b><i> (<var>expr</var>)</i>
<a name="IDX123"></a>
</dt>
<dt><u>Function:</u> <b>box</b><i> (<var>expr</var>, <var>a</var>)</i>
<a name="IDX124"></a>
</dt>
<dd><p>Returns <var>expr</var> enclosed in a box.
The return value is an expression with <code>box</code> as the operator and <var>expr</var> as the argument.
A box is drawn on the display when <code>display2d</code> is <code>true</code>.
</p>
<p><code>box (<var>expr</var>, <var>a</var>)</code>
encloses <var>expr</var> in a box labelled by the symbol <var>a</var>.
The label is truncated if it is longer than the width of the box.
</p>
<p><code>box</code> evaluates its argument.
However, a boxed expression does not evaluate to its content,
so boxed expressions are effectively excluded from computations.
</p>
<p><code>boxchar</code> is the character used to draw the box in <code>box</code>
and in the <code>dpart</code> and <code>lpart</code> functions.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) box (a^2 + b^2);
                            &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                            &quot; 2    2&quot;
(%o1)                       &quot;b  + a &quot;
                            &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
(%i2) a : 1234;
(%o2)                         1234
(%i3) b : c - d;
(%o3)                         c - d
(%i4) box (a^2 + b^2);
                      &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                      &quot;       2          &quot;
(%o4)                 &quot;(c - d)  + 1522756&quot;
                      &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
(%i5) box (a^2 + b^2, term_1);
                      term_1&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                      &quot;       2          &quot;
(%o5)                 &quot;(c - d)  + 1522756&quot;
                      &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
(%i6) 1729 - box (1729);
                                 &quot;&quot;&quot;&quot;&quot;&quot;
(%o6)                     1729 - &quot;1729&quot;
                                 &quot;&quot;&quot;&quot;&quot;&quot;
(%i7) boxchar: &quot;-&quot;;
(%o7)                           -
(%i8) box (sin(x) + cos(y));
                        -----------------
(%o8)                   -cos(y) + sin(x)-
                        -----------------
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>boxchar</b>
<a name="IDX125"></a>
</dt>
<dd><p>Default value: <code>&quot;</code>
</p>
<p><code>boxchar</code> is the character used to draw the box in the <code>box</code>
and in the <code>dpart</code> and <code>lpart</code> functions.
</p>
<p>All boxes in an expression are drawn with the current value of <code>boxchar</code>;
the drawing character is not stored with the box expression.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>carg</b><i> (<var>z</var>)</i>
<a name="IDX126"></a>
</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><code>carg</code> ignores the declaration <code>declare (<var>x</var>, complex)</code>,
and treats <var>x</var> as a real variable.
This is a bug. </p>
<p>See also <code>abs</code> (complex magnitude), <code>polarform</code>, <code>rectform</code>,
<code>realpart</code>, and <code>imagpart</code>.
</p>
<p>Examples:
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) carg (1);
(%o1)                           0
(%i2) carg (1 + %i);
                               %pi
(%o2)                          ---
                                4
(%i3) carg (exp (%i));
(%o3)                           1
(%i4) carg (exp (%pi * %i));
(%o4)                          %pi
(%i5) carg (exp (3/2 * %pi * %i));
                                %pi
(%o5)                         - ---
                                 2
(%i6) carg (17 * exp (2 * %i));
(%o6)                           2
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Special operator:</u> <b>constant</b>
<a name="IDX127"></a>
</dt>
<dd><p><code>declare (<var>a</var>, constant)</code> declares <var>a</var> to be a constant.
See <code>declare</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>constantp</b><i> (<var>expr</var>)</i>
<a name="IDX128"></a>
</dt>
<dd><p>Returns <code>true</code> if <var>expr</var> is a constant expression,
otherwise returns <code>false</code>.
</p>
<p>An expression is considered a constant expression if its arguments are
numbers (including rational numbers, as displayed with <code>/R/</code>),
symbolic constants such as <code>%pi</code>, <code>%e</code>, and <code>%i</code>,
variables bound to a constant or declared constant by <code>declare</code>,
or functions whose arguments are constant.
</p>
<p><code>constantp</code> evaluates its arguments.
</p>
<p>Examples:
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) constantp (7 * sin(2));
(%o1)                                true
(%i2) constantp (rat (17/29));
(%o2)                                true
(%i3) constantp (%pi * sin(%e));
(%o3)                                true
(%i4) constantp (exp (x));
(%o4)                                false
(%i5) declare (x, constant);
(%o5)                                done
(%i6) constantp (exp (x));
(%o6)                                true
(%i7) constantp (foo (x) + bar (%e) + baz (2));
(%o7)                                false
(%i8) 
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>declare</b><i> (<var>a_1</var>, <var>p_1</var>, <var>a_2</var>, <var>p_2</var>, ...)</i>
<a name="IDX129"></a>
</dt>
<dd><p>Assigns the atom or list of atoms <var>a_i</var> the property or list of properties <var>p_i</var>.
When <var>a_i</var> and/or <var>p_i</var> are lists,
each of the atoms gets all of the properties.
</p>
<p><code>declare</code> quotes its arguments.
<code>declare</code> always returns <code>done</code>.
</p>

<p>As noted in the description for each declaration flag,
for some flags
<code>featurep(<var>object</var>, <var>feature</var>)</code>
returns <code>true</code> if <var>object</var> has been declared to have <var>feature</var>.
However, <code>featurep</code> does not recognize some flags; this is a bug.
</p>
<p>See also <code>features</code>.
</p>
<p><code>declare</code> recognizes the following properties:
</p>
<dl compact="compact">
<dt> <code>evfun</code></dt>
<dd><p>Makes <var>a_i</var> known to <code>ev</code> so that the function named by <var>a_i</var>
is applied when <var>a_i</var> appears as a flag argument of <code>ev</code>.
See <code>evfun</code>.
</p>
</dd>
<dt> <code>evflag</code></dt>
<dd><p>Makes <var>a_i</var> known to the <code>ev</code> function so that <var>a_i</var> is bound to <code>true</code>
during the execution of <code>ev</code> when <var>a_i</var> appears as a flag argument of <code>ev</code>.
See <code>evflag</code>.
</p>


</dd>
<dt> <code>bindtest</code></dt>
<dd><p>Tells Maxima to trigger an error when <var>a_i</var> is evaluated unbound.
</p>
</dd>
<dt> <code>noun</code></dt>
<dd><p>Tells Maxima to parse <var>a_i</var> as a noun. 
The effect of this is to replace instances of <var>a_i</var> with <code>'<var>a_i</var></code>
or <code>nounify(<var>a_i</var>)</code>, depending on the context.
</p>
</dd>
<dt> <code>constant</code></dt>
<dd><p>Tells Maxima to consider <var>a_i</var> a symbolic constant.
</p>
</dd>
<dt> <code>scalar</code></dt>
<dd><p>Tells Maxima to consider <var>a_i</var> a scalar variable.
</p>
</dd>
<dt> <code>nonscalar</code></dt>
<dd><p>Tells Maxima to consider <var>a_i</var> a nonscalar variable.
The usual application is to declare a variable as a symbolic vector or matrix.
</p>
</dd>
<dt> <code>mainvar</code></dt>
<dd><p>Tells Maxima to consider <var>a_i</var> a &quot;main variable&quot; (<code>mainvar</code>).
<code>ordergreatp</code> determines the ordering of atoms as follows:
</p>
<p>(main variables) &gt; (other variables) &gt; (scalar variables) &gt; (constants) &gt; (numbers)
</p>
</dd>
<dt> <code>alphabetic</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an alphabetic character.
</p>
</dd>
<dt> <code>feature</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as the name of a feature.
Other atoms may then be declared to have the <var>a_i</var> property.
</p>
</dd>
<dt> <code>rassociative</code>, <code>lassociative</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as a right-associative or left-associative function.
</p>
</dd>
<dt> <code>nary</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an n-ary function.
</p>
<p>The <code>nary</code> declaration is not the same as calling the <code>nary</code> function.
The sole effect of <code>declare(foo, nary)</code> is to instruct the Maxima simplifier
to flatten nested expressions,
for example, to simplify <code>foo(x, foo(y, z))</code> to <code>foo(x, y, z)</code>.
</p>
</dd>
<dt> <code>symmetric</code>, <code>antisymmetric</code>, <code>commutative</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as a symmetric or antisymmetric function.
<code>commutative</code> is the same as <code>symmetric</code>.
</p>
</dd>
<dt> <code>oddfun</code>, <code>evenfun</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an odd or even function.
</p>
</dd>
<dt> <code>outative</code></dt>
<dd><p>Tells Maxima to simplify <var>a_i</var> expressions
by pulling constant factors out of the first argument.
</p>
<p>When <var>a_i</var> has one argument,
a factor is considered constant if it is a literal or declared constant.
</p>
<p>When <var>a_i</var> has two or more arguments,
a factor is considered constant
if the second argument is a symbol
and the factor is free of the second argument.
</p>
</dd>
<dt> <code>multiplicative</code></dt>
<dd><p>Tells Maxima to simplify <var>a_i</var> expressions
by the substitution <code><var>a_i</var>(x * y * z * ...)</code> <code>--&gt;</code>
<code><var>a_i</var>(x) * <var>a_i</var>(y) * <var>a_i</var>(z) * ...</code>.
The substitution is carried out on the first argument only.
</p>
</dd>
<dt> <code>additive</code></dt>
<dd><p>Tells Maxima to simplify <var>a_i</var> expressions
by the substitution <code><var>a_i</var>(x + y + z + ...)</code> <code>--&gt;</code>
<code><var>a_i</var>(x) + <var>a_i</var>(y) + <var>a_i</var>(z) + ...</code>.
The substitution is carried out on the first argument only.
</p>
</dd>
<dt> <code>linear</code></dt>
<dd><p>Equivalent to declaring <var>a_i</var> both <code>outative</code> and <code>additive</code>.
</p>

</dd>
<dt> <code>integer</code>, <code>noninteger</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an integer or noninteger variable.
</p>
</dd>
<dt> <code>even</code>, <code>odd</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an even or odd integer variable.
</p>
</dd>
<dt> <code>rational</code>, <code>irrational</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as a rational or irrational real variable.
</p>
</dd>
<dt> <code>real</code>, <code>imaginary</code>, <code>complex</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as a real, pure imaginary, or complex variable.
</p>
</dd>
<dt> <code>increasing</code>, <code>decreasing</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an increasing or decreasing function.
</p>
</dd>
<dt> <code>posfun</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as a positive function.
</p>
</dd>
<dt> <code>integervalued</code></dt>
<dd><p>Tells Maxima to recognize <var>a_i</var> as an integer-valued function.
</p>
</dd>
</dl>

<p>Examples:
</p>
<p><code>evfun</code> and <code>evflag</code> declarations.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) declare (expand, evfun);
(%o1)                         done
(%i2) (a + b)^3;
                                   3
(%o2)                       (b + a)
(%i3) (a + b)^3, expand;
                     3        2      2      3
(%o3)               b  + 3 a b  + 3 a  b + a
(%i4) declare (demoivre, evflag);
(%o4)                         done
(%i5) exp (a + b*%i);
                             %i b + a
(%o5)                      %e
(%i6) exp (a + b*%i), demoivre;
                      a
(%o6)               %e  (%i sin(b) + cos(b))
</pre></td></tr></table>
<p><code>bindtest</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) aa + bb;
(%o1)                        bb + aa
(%i2) declare (aa, bindtest);
(%o2)                         done
(%i3) aa + bb;
aa unbound variable
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i4) aa : 1234;
(%o4)                         1234
(%i5) aa + bb;
(%o5)                       bb + 1234
</pre></td></tr></table>
<p><code>noun</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) factor (12345678);
                             2
(%o1)                     2 3  47 14593
(%i2) declare (factor, noun);
(%o2)                         done
(%i3) factor (12345678);
(%o3)                   factor(12345678)
(%i4) ''%, nouns;
                             2
(%o4)                     2 3  47 14593
</pre></td></tr></table>
<p><code>constant</code>, <code>scalar</code>, <code>nonscalar</code>, and <code>mainvar</code> declarations.
</p>
<p><code>alphabetic</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) xx\~yy : 1729;
(%o1)                         1729
(%i2) declare (&quot;~&quot;, alphabetic);
(%o2)                         done
(%i3) xx~yy + yy~xx + ~xx~~yy~;
(%o3)                ~xx~~yy~ + yy~xx + 1729
</pre></td></tr></table>
<p><code>feature</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) declare (FOO, feature);
(%o1)                         done
(%i2) declare (x, FOO);
(%o2)                         done
(%i3) featurep (x, FOO);
(%o3)                         true
</pre></td></tr></table>
<p><code>rassociative</code> and <code>lassociative</code> declarations.
</p>
<p><code>nary</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) H (H (a, b), H (c, H (d, e)));
(%o1)               H(H(a, b), H(c, H(d, e)))
(%i2) declare (H, nary);
(%o2)                         done
(%i3) H (H (a, b), H (c, H (d, e)));
(%o3)                   H(a, b, c, d, e)
</pre></td></tr></table>
<p><code>symmetric</code> and <code>antisymmetric</code> declarations.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) S (b, a);
(%o1)                        S(b, a)
(%i2) declare (S, symmetric);
(%o2)                         done
(%i3) S (b, a);
(%o3)                        S(a, b)
(%i4) S (a, c, e, d, b);
(%o4)                   S(a, b, c, d, e)
(%i5) T (b, a);
(%o5)                        T(b, a)
(%i6) declare (T, antisymmetric);
(%o6)                         done
(%i7) T (b, a);
(%o7)                       - T(a, b)
(%i8) T (a, c, e, d, b);
(%o8)                   T(a, b, c, d, e)
</pre></td></tr></table>
<p><code>oddfun</code> and <code>evenfun</code> declarations.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) o (- u) + o (u);
(%o1)                     o(u) + o(- u)
(%i2) declare (o, oddfun);
(%o2)                         done
(%i3) o (- u) + o (u);
(%o3)                           0
(%i4) e (- u) - e (u);
(%o4)                     e(- u) - e(u)
(%i5) declare (e, evenfun);
(%o5)                         done
(%i6) e (- u) - e (u);
(%o6)                           0
</pre></td></tr></table>
<p><code>outative</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) F1 (100 * x);
(%o1)                       F1(100 x)
(%i2) declare (F1, outative);
(%o2)                         done
(%i3) F1 (100 * x);
(%o3)                       100 F1(x)
(%i4) declare (zz, constant);
(%o4)                         done
(%i5) F1 (zz * y);
(%o5)                       zz F1(y)
</pre></td></tr></table>
<p><code>multiplicative</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) F2 (a * b * c);
(%o1)                       F2(a b c)
(%i2) declare (F2, multiplicative);
(%o2)                         done
(%i3) F2 (a * b * c);
(%o3)                   F2(a) F2(b) F2(c)
</pre></td></tr></table>
<p><code>additive</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) F3 (a + b + c);
(%o1)                     F3(c + b + a)
(%i2) declare (F3, additive);
(%o2)                         done
(%i3) F3 (a + b + c);
(%o3)                 F3(c) + F3(b) + F3(a)
</pre></td></tr></table>
<p><code>linear</code> declaration.
</p><table><tr><td>&nbsp;</td><td><pre class="example">(%i1) 'sum (F(k) + G(k), k, 1, inf);
                       inf
                       ====
                       \
(%o1)                   &gt;    (G(k) + F(k))
                       /
                       ====
                       k = 1
(%i2) declare (nounify (sum), linear);
(%o2)                         done
(%i3) 'sum (F(k) + G(k), k, 1, inf);
                     inf          inf
                     ====         ====
                     \            \
(%o3)                 &gt;    G(k) +  &gt;    F(k)
                     /            /
                     ====         ====
                     k = 1        k = 1
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>disolate</b><i> (<var>expr</var>, <var>x_1</var>, ..., <var>x_n</var>)</i>
<a name="IDX130"></a>
</dt>
<dd><p>is similar to <code>isolate (<var>expr</var>, <var>x</var>)</code>
except that it enables the user to isolate
more than one variable simultaneously.  This might be useful, for
example, if one were attempting to change variables in a multiple
integration, and that variable change involved two or more of the
integration variables.  This function is autoloaded from
<tt>`simplification/disol.mac'</tt>.  A demo is available by
<code>demo(&quot;disol&quot;)$</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>dispform</b><i> (<var>expr</var>)</i>
<a name="IDX131"></a>
</dt>
<dd><p>Returns the external representation of <var>expr</var> with respect to its
main operator.  This should be useful in conjunction with <code>part</code> which
also deals with the external representation.  Suppose <var>expr</var> is -A .
Then the internal representation of <var>expr</var> is &quot;*&quot;(-1,A), while the
external representation is &quot;-&quot;(A). <code>dispform (<var>expr</var>, all)</code> converts the
entire expression (not just the top-level) to external format.  For
example, if <code>expr: sin (sqrt (x))</code>, then <code>freeof (sqrt, expr)</code> and
<code>freeof (sqrt, dispform (expr))</code> give <code>true</code>, while
<code>freeof (sqrt, dispform (expr, all))</code> gives <code>false</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>distrib</b><i> (<var>expr</var>)</i>
<a name="IDX132"></a>
</dt>
<dd><p>Distributes sums over products.  It differs from <code>expand</code>
in that it works at only the top level of an expression, i.e., it doesn't
recurse and it is faster than <code>expand</code>.  It differs from <code>multthru</code> in
that it expands all sums at that level.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) distrib ((a+b) * (c+d));
(%o1)                 b d + a d + b c + a c
(%i2) multthru ((a+b) * (c+d));
(%o2)                 (b + a) d + (b + a) c
(%i3) distrib (1/((a+b) * (c+d)));
                                1
(%o3)                    ---------------
                         (b + a) (d + c)
(%i4) expand (1/((a+b) * (c+d)), 1, 0);
                                1
(%o4)                 ---------------------
                      b d + a d + b c + a c
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>dpart</b><i> (<var>expr</var>, <var>n_1</var>, ..., <var>n_k</var>)</i>
<a name="IDX133"></a>
</dt>
<dd><p>Selects the same subexpression as <code>part</code>, but
instead of just returning that subexpression as its value, it returns
the whole expression with the selected subexpression displayed inside
a box.  The box is actually part of the expression.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) dpart (x+y/z^2, 1, 2, 1);
                             y
(%o1)                       ---- + x
                               2
                            &quot;&quot;&quot;
                            &quot;z&quot;
                            &quot;&quot;&quot;
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>exp</b><i> (<var>x</var>)</i>
<a name="IDX134"></a>
</dt>
<dd><p>Represents the exponential function.  
Instances of <code>exp (<var>x</var>)</code> in input are simplified to <code>%e^<var>x</var></code>;
<code>exp</code> does not appear in simplified expressions.
</p>
<p><code>demoivre</code> if <code>true</code> causes <code>%e^(a + b %i)</code> to simplify to
<code>%e^(a (cos(b) + %i sin(b)))</code> if <code>b</code> is free of <code>%i</code>. See <code>demoivre</code>.
</p>
<p><code>%emode</code>, when <code>true</code>, 
causes <code>%e^(%pi %i x)</code> to be simplified. See <code>%emode</code>.
</p>
<p><code>%enumer</code>, when <code>true</code> causes <code>%e</code> to be replaced by
2.718...  whenever <code>numer</code> is <code>true</code>. See <code>%enumer</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>%emode</b>
<a name="IDX135"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>%emode</code> is <code>true</code>,
<code>%e^(%pi %i x)</code> is simplified as
follows.
</p>
<p><code>%e^(%pi %i x)</code> simplifies to <code>cos (%pi x) + %i sin (%pi x)</code> if <code>x</code> is an integer or
a multiple of 1/2, 1/3, 1/4, or 1/6, and then further simplified.
</p>
<p>For other numerical <code>x</code>,
<code>%e^(%pi %i x)</code> simplifies to <code>%e^(%pi %i y)</code> where <code>y</code> is <code>x - 2 k</code>
for some integer <code>k</code> such that <code>abs(y) &lt; 1</code>.  
</p>
<p>When <code>%emode</code> is <code>false</code>, no
special simplification of <code>%e^(%pi %i x)</code> is carried out.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>%enumer</b>
<a name="IDX136"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>%enumer</code> is <code>true</code>,
<code>%e</code> is replaced by its numeric value
2.718...  whenever <code>numer</code> is <code>true</code>. 
</p>
<p>When <code>%enumer</code> is <code>false</code>, this substitution is carried out
only if the exponent in <code>%e^x</code> evaluates to a number.
</p>
<p>See also <code>ev</code> and <code>numer</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>exptisolate</b>
<a name="IDX137"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>exptisolate</code>, when <code>true</code>, causes <code>isolate (expr, var)</code> to
examine exponents of atoms (such as <code>%e</code>) which contain <code>var</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>exptsubst</b>
<a name="IDX138"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>exptsubst</code>, when <code>true</code>, permits substitutions such as <code>y</code>
for <code>%e^x</code> in <code>%e^(a x)</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>freeof</b><i> (<var>x_1</var>, ..., <var>x_n</var>, <var>expr</var>)</i>
<a name="IDX139"></a>
</dt>
<dd><p><code>freeof (<var>x_1</var>, <var>expr</var>)</code>
Returns <code>true</code>
if no subexpression of <var>expr</var> is equal to <var>x_1</var>
or if <var>x_1</var> occurs only as a dummy variable in <var>expr</var>,
and returns <code>false</code> otherwise.
</p>
<p><code>freeof (<var>x_1</var>, ..., <var>x_n</var>, <var>expr</var>)</code>
is equivalent to <code>freeof (<var>x_1</var>, <var>expr</var>) and ... and freeof (<var>x_n</var>, <var>expr</var>)</code>.
</p>
<p>The arguments <var>x_1</var>, ..., <var>x_n</var> 
may be names of functions and variables, subscripted names,
operators (enclosed in double quotes), or general expressions.
<code>freeof</code> evaluates its arguments.
</p>
<p><code>freeof</code> operates only on <var>expr</var> as it stands (after simplification and evaluation) and
does not attempt to determine if some equivalent expression would give a different result.
In particular, simplification may yield an equivalent but different expression which comprises
some different elements than the original form of <var>expr</var>.
</p>
<p>A variable is a dummy variable in an expression if it has no binding outside of the expression.
Dummy variables recognized by <code>freeof</code> are
the index of a sum or product, the limit variable in <code>limit</code>,
the integration variable in the definite integral form of <code>integrate</code>,
the original variable in <code>laplace</code>,
formal variables in <code>at</code> expressions,
and arguments in <code>lambda</code> expressions.
Local variables in <code>block</code> are not recognized by <code>freeof</code> as dummy variables;
this is a bug.
</p>
<p>The indefinite form of <code>integrate</code> is <i>not</i> free of its variable of integration.
</p>
<ul>
<li>
Arguments are names of functions, variables, subscripted names, operators, and expressions.
<code>freeof (a, b, expr)</code> is equivalent to
<code>freeof (a, expr) and freeof (b, expr)</code>.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: z^3 * cos (a[1]) * b^(c+d);
                                 d + c  3
(%o1)                   cos(a ) b      z
                             1
(%i2) freeof (z, expr);
(%o2)                         false
(%i3) freeof (cos, expr);
(%o3)                         false
(%i4) freeof (a[1], expr);
(%o4)                         false
(%i5) freeof (cos (a[1]), expr);
(%o5)                         false
(%i6) freeof (b^(c+d), expr);
(%o6)                         false
(%i7) freeof (&quot;^&quot;, expr);
(%o7)                         false
(%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr);
(%o8)                         true
</pre></td></tr></table>
</li><li>
<code>freeof</code> evaluates its arguments.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: (a+b)^5$
(%i2) c: a$
(%i3) freeof (c, expr);
(%o3)                         false
</pre></td></tr></table>
</li><li>
<code>freeof</code> does not consider equivalent expressions.
Simplification may yield an equivalent but different expression.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: (a+b)^5$
(%i2) expand (expr);
          5        4       2  3       3  2      4      5
(%o2)    b  + 5 a b  + 10 a  b  + 10 a  b  + 5 a  b + a
(%i3) freeof (a+b, %);
(%o3)                         true
(%i4) freeof (a+b, expr);
(%o4)                         false
(%i5) exp (x);
                                 x
(%o5)                          %e
(%i6) freeof (exp, exp (x));
(%o6)                         true
</pre></td></tr></table>
</li><li> A summation or definite integral is free of its dummy variable.
An indefinite integral is not free of its variable of integration.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) freeof (i, 'sum (f(i), i, 0, n));
(%o1)                         true
(%i2) freeof (x, 'integrate (x^2, x, 0, 1));
(%o2)                         true
(%i3) freeof (x, 'integrate (x^2, x));
(%o3)                         false
</pre></td></tr></table></li></ul>

</dd></dl>

<dl>
<dt><u>Function:</u> <b>genfact</b><i> (<var>x</var>, <var>y</var>, <var>z</var>)</i>
<a name="IDX140"></a>
</dt>
<dd><p>Returns the generalized factorial, defined as
<code>x (x-z) (x - 2 z) ... (x - (y - 1) z)</code>.  Thus, for integral <var>x</var>,
<code>genfact (x, x, 1) = x!</code> and <code>genfact (x, x/2, 2) = x!!</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>imagpart</b><i> (<var>expr</var>)</i>
<a name="IDX141"></a>
</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>abs</code>, <code>carg</code>, <code>polarform</code>, <code>rectform</code>,
and <code>realpart</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>infix</b><i> (<var>op</var>)</i>
<a name="IDX142"></a>
</dt>
<dt><u>Function:</u> <b>infix</b><i> (<var>op</var>, <var>lbp</var>, <var>rbp</var>)</i>
<a name="IDX143"></a>
</dt>
<dt><u>Function:</u> <b>infix</b><i> (<var>op</var>, <var>lbp</var>, <var>rbp</var>, <var>lpos</var>, <var>rpos</var>, <var>pos</var>)</i>
<a name="IDX144"></a>
</dt>
<dd><p>Declares <var>op</var> to be an infix operator.
An infix operator is a function of two arguments,
with the name of the function written between the arguments.
For example, the subtraction operator <code>-</code> is an infix operator.
</p>
<p><code>infix (<var>op</var>)</code> declares <var>op</var> to be an infix operator
with default binding powers (left and right both equal to 180)
and parts of speech (left and right both equal to <code>any</code>).
</p>
<p><code>infix (<var>op</var>, <var>lbp</var>, <var>rbp</var>)</code> declares <var>op</var> to be an infix operator
with stated left and right binding powers
and default parts of speech (left and right both equal to <code>any</code>).
</p>
<p><code>infix (<var>op</var>, <var>lbp</var>, <var>rbp</var>, <var>lpos</var>, <var>rpos</var>, <var>pos</var>)</code>
declares <var>op</var> to be an infix operator
with stated left and right binding powers and parts of speech.
</p>
<p>The precedence of <var>op</var> with respect to other operators
derives from the left and right binding powers of the operators in question.
If the left and right binding powers of <var>op</var> are both greater
the left and right binding powers of some other operator,
then <var>op</var> takes precedence over the other operator.
If the binding powers are not both greater or less,
some more complicated relation holds.
</p>
<p>The associativity of <var>op</var> depends on its binding powers.
Greater left binding power (<var>lbp</var>) implies an instance of
<var>op</var> is evaluated before other operators to its left in an expression,
while greater right binding power (<var>rbp</var>) implies  an instance of
<var>op</var> is evaluated before other operators to its right in an expression.
Thus greater <var>lbp</var> makes <var>op</var> right-associative,
while greater <var>rbp</var> makes <var>op</var> left-associative.
If <var>lbp</var> is equal to <var>rbp</var>, <var>op</var> is left-associative.
</p>
<p>See also <code>Syntax</code>.
</p>
<p>Examples:
</p>
<ul>
<li>
If the left and right binding powers of <var>op</var> are both greater
the left and right binding powers of some other operator,
then <var>op</var> takes precedence over the other operator.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) &quot;@&quot;(a, b) := sconcat(&quot;(&quot;, a, &quot;,&quot;, b, &quot;)&quot;)$
(%i2) :lisp (get '$+ 'lbp)
100
(%i2) :lisp (get '$+ 'rbp)
100
(%i2) infix (&quot;@&quot;, 101, 101)$
(%i3) 1 + a@b + 2;
(%o3)                       (a,b) + 3
(%i4) infix (&quot;@&quot;, 99, 99)$
(%i5) 1 + a@b + 2;
(%o5)                       (a+1,b+2)
</pre></td></tr></table>
<ul>
<li>
Greater <var>lbp</var> makes <var>op</var> right-associative,
while greater <var>rbp</var> makes <var>op</var> left-associative.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) &quot;@&quot;(a, b) := sconcat(&quot;(&quot;, a, &quot;,&quot;, b, &quot;)&quot;)$
(%i2) infix (&quot;@&quot;, 100, 99)$
(%i3) foo @ bar @ baz;
(%o3)                    (foo,(bar,baz))
(%i4) infix (&quot;@&quot;, 100, 101)$
(%i5) foo @ bar @ baz;
(%o5)                    ((foo,bar),baz)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>inflag</b>
<a name="IDX145"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>inflag</code> is <code>true</code>, functions for part
extraction inspect the internal form of <code>expr</code>.
</p>
<p>Note that the simplifier re-orders expressions.
Thus <code>first (x + y)</code> returns <code>x</code> if <code>inflag</code>
is <code>true</code> and <code>y</code> if <code>inflag</code> is <code>false</code>.
(<code>first (y + x)</code> gives the same results.)
</p>
<p>Also, setting <code>inflag</code> to <code>true</code> and calling <code>part</code> or <code>substpart</code> is
the same as calling <code>inpart</code> or <code>substinpart</code>.
</p>
<p>Functions affected by the setting of <code>inflag</code> are:
<code>part</code>, <code>substpart</code>, <code>first</code>, <code>rest</code>, <code>last</code>, <code>length</code>,
the <code>for</code> ... <code>in</code> construct,
<code>map</code>, <code>fullmap</code>, <code>maplist</code>, <code>reveal</code> and <code>pickapart</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>inpart</b><i> (<var>expr</var>, <var>n_1</var>, ..., <var>n_k</var>)</i>
<a name="IDX146"></a>
</dt>
<dd><p>is similar to <code>part</code> but works on the internal
representation of the expression rather than the displayed form and
thus may be faster since no formatting is done.  Care should be taken
with respect to the order of subexpressions in sums and products
(since the order of variables in the internal form is often different
from that in the displayed form) and in dealing with unary minus,
subtraction, and division (since these operators are removed from the
expression). <code>part (x+y, 0)</code> or <code>inpart (x+y, 0)</code> yield <code>+</code>, though in order to
refer to the operator it must be enclosed in &quot;s.  For example
<code>... if inpart (%o9,0) = &quot;+&quot; then ...</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) x + y + w*z;
(%o1)                      w z + y + x
(%i2) inpart (%, 3, 2);
(%o2)                           z
(%i3) part (%th (2), 1, 2);
(%o3)                           z
(%i4) 'limit (f(x)^g(x+1), x, 0, minus);
                                  g(x + 1)
(%o4)                 limit   f(x)
                      x -&gt; 0-
(%i5) inpart (%, 1, 2);
(%o5)                       g(x + 1)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>isolate</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX147"></a>
</dt>
<dd><p>Returns <var>expr</var> with subexpressions which are sums and
which do not contain var replaced by intermediate expression labels
(these being atomic symbols like <code>%t1</code>, <code>%t2</code>, ...).  This is often useful
to avoid unnecessary expansion of subexpressions which don't contain
the variable of interest.  Since the intermediate labels are bound to
the subexpressions they can all be substituted back by evaluating the
expression in which they occur.
</p>
<p><code>exptisolate</code> (default value: <code>false</code>) if <code>true</code> will cause <code>isolate</code> to examine exponents of
atoms (like <code>%e</code>) which contain var.
</p>
<p><code>isolate_wrt_times</code> if <code>true</code>, then <code>isolate</code> will also isolate with respect to
products. See <code>isolate_wrt_times</code>.
</p>
<p>Do <code>example (isolate)</code> for examples.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>isolate_wrt_times</b>
<a name="IDX148"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>isolate_wrt_times</code> is <code>true</code>, <code>isolate</code>
will also isolate with respect to products.  E.g. compare both settings of the
switch on
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) isolate_wrt_times: true$
(%i2) isolate (expand ((a+b+c)^2), c);

(%t2)                          2 a


(%t3)                          2 b


                          2            2
(%t4)                    b  + 2 a b + a

                     2
(%o4)               c  + %t3 c + %t2 c + %t4
(%i4) isolate_wrt_times: false$
(%i5) isolate (expand ((a+b+c)^2), c);
                     2
(%o5)               c  + 2 b c + 2 a c + %t4
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>listconstvars</b>
<a name="IDX149"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>listconstvars</code> is <code>true</code>, it will cause <code>listofvars</code> to
include <code>%e</code>, <code>%pi</code>, <code>%i</code>, and any variables declared constant in the list
it returns if they appear in the expression <code>listofvars</code> is called on.
The default is to omit these.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>listdummyvars</b>
<a name="IDX150"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>listdummyvars</code> is <code>false</code>, &quot;dummy variables&quot; in the
expression will not be included in the list returned by <code>listofvars</code>.
(The meaning of &quot;dummy variables&quot; is as given in <code>freeof</code>.
&quot;Dummy variables&quot; are mathematical things like the index of a sum or
product, the limit variable, and the definite integration variable.)
Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) listdummyvars: true$
(%i2) listofvars ('sum(f(i), i, 0, n));
(%o2)                        [i, n]
(%i3) listdummyvars: false$
(%i4) listofvars ('sum(f(i), i, 0, n));
(%o4)                          [n]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>listofvars</b><i> (<var>expr</var>)</i>
<a name="IDX151"></a>
</dt>
<dd><p>Returns a list of the variables in <var>expr</var>.
</p>
<p><code>listconstvars</code> if <code>true</code> causes <code>listofvars</code> to include <code>%e</code>, <code>%pi</code>,
<code>%i</code>, and any variables declared constant in the list it returns if they
appear in <var>expr</var>.  The default is to omit these.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) listofvars (f (x[1]+y) / g^(2+a));
(%o1)                     [g, a, x , y]
                                  1
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>lfreeof</b><i> (<var>list</var>, <var>expr</var>)</i>
<a name="IDX152"></a>
</dt>
<dd><p>For each member <var>m</var> of list, calls <code>freeof (<var>m</var>, <var>expr</var>)</code>.
It returns <code>false</code> if any call to <code>freeof</code> does and <code>true</code> otherwise.
</p></dd></dl>

<dl>
<dt><u>Function:</u> <b>lopow</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX153"></a>
</dt>
<dd><p>Returns the lowest exponent of <var>x</var> which explicitly appears in
<var>expr</var>.  Thus
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
(%o1)                       min(a, 2)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>lpart</b><i> (<var>label</var>, <var>expr</var>, <var>n_1</var>, ..., <var>n_k</var>)</i>
<a name="IDX154"></a>
</dt>
<dd><p>is similar to <code>dpart</code> but uses a
labelled box. A labelled box is similar to the one produced by <code>dpart</code>
but it has a name in the top line.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>multthru</b><i> (<var>expr</var>)</i>
<a name="IDX155"></a>
</dt>
<dt><u>Function:</u> <b>multthru</b><i> (<var>expr_1</var>, <var>expr_2</var>)</i>
<a name="IDX156"></a>
</dt>
<dd><p>Multiplies a factor (which should be a sum) of <var>expr</var> by
the other factors of <var>expr</var>.  That is, <var>expr</var> is <code><var>f_1</var> <var>f_2</var> ... <var>f_n</var></code>
where at least
one factor, say <var>f_i</var>, is a sum of terms.  Each term in that sum is
multiplied by the other factors in the product.  (Namely all the
factors except <var>f_i</var>).  <code>multthru</code> does not expand exponentiated sums.
This function is the fastest way to distribute products (commutative
or noncommutative) over sums.  Since quotients are represented as
products <code>multthru</code> can be used to divide sums by products as well.
</p>
<p><code>multthru (<var>expr_1</var>, <var>expr_2</var>)</code> multiplies each term in <var>expr_2</var> (which should be a
sum or an equation) by <var>expr_1</var>.  If <var>expr_1</var> is not itself a sum then this
form is equivalent to <code>multthru (<var>expr_1</var>*<var>expr_2</var>)</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
                      1        x         f(x)
(%o1)             - ----- + -------- - --------
                    x - y          2          3
                            (x - y)    (x - y)
(%i2) multthru ((x-y)^3, %);
                           2
(%o2)             - (x - y)  + x (x - y) - f(x)
(%i3) ratexpand (%);
                           2
(%o3)                   - y  + x y - f(x)
(%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
                        10  2              2  2
                 (b + a)   s  + 2 a b s + a  b
(%o4)            ------------------------------
                                  2
                             a b s
(%i5) multthru (%);  /* note that this does not expand (b+a)^10 */
                                        10
                       2   a b   (b + a)
(%o5)                  - + --- + ---------
                       s    2       a b
                           s
(%i6) multthru (a.(b+c.(d+e)+f));
(%o6)            a . f + a . c . (e + d) + a . b
(%i7) expand (a.(b+c.(d+e)+f));
(%o7)         a . f + a . c . e + a . c . d + a . b
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>nounify</b><i> (<var>f</var>)</i>
<a name="IDX157"></a>
</dt>
<dd><p>Returns the noun form of the function name <var>f</var>.  This is
needed if one wishes to refer to the name of a verb function as if it
were a noun.  Note that some verb functions will return their noun
forms if they can't be evaluated for certain arguments.  This is also
the form returned if a function call is preceded by a quote.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>nterms</b><i> (<var>expr</var>)</i>
<a name="IDX158"></a>
</dt>
<dd><p>Returns the number of terms that <var>expr</var> would have if it were
fully expanded out and no cancellations or combination of terms
occurred.
Note that expressions like <code>sin (<var>expr</var>)</code>, <code>sqrt (<var>expr</var>)</code>, <code>exp (<var>expr</var>)</code>, etc.
count as just one term regardless of how many terms <var>expr</var> has (if it is a
sum).
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>op</b><i> (<var>expr</var>)</i>
<a name="IDX159"></a>
</dt>
<dd><p>Returns the main operator of the expression <var>expr</var>.
<code>op (<var>expr</var>)</code> is equivalent to <code>part (<var>expr</var>, 0)</code>. 
</p>
<p><code>op</code> returns a string if the main operator is
a built-in or user-defined
prefix, binary or n-ary infix, postfix, matchfix, or nofix operator.
Otherwise <code>op</code> returns a symbol.
</p>
<p><code>op</code> observes the value of the global flag <code>inflag</code>.
</p>
<p><code>op</code> evaluates it argument.
</p>
<p>See also <code>args</code>.
</p>
<p>Examples:
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) ?stringdisp: true$
(%i2) op (a * b * c);
(%o2)                          &quot;*&quot;
(%i3) op (a * b + c);
(%o3)                          &quot;+&quot;
(%i4) op ('sin (a + b));
(%o4)                          sin
(%i5) op (a!);
(%o5)                          &quot;!&quot;
(%i6) op (-a);
(%o6)                          &quot;-&quot;
(%i7) op ([a, b, c]);
(%o7)                          &quot;[&quot;
(%i8) op ('(if a &gt; b then c else d));
(%o8)                         &quot;if&quot;
(%i9) op ('foo (a));
(%o9)                          foo
(%i10) prefix (foo);
(%o10)                        &quot;foo&quot;
(%i11) op (foo a);
(%o11)                        &quot;foo&quot;
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>operatorp</b><i> (<var>expr</var>, <var>op</var>)</i>
<a name="IDX160"></a>
</dt>
<dt><u>Function:</u> <b>operatorp</b><i> (<var>expr</var>, [<var>op_1</var>, ..., <var>op_n</var>])</i>
<a name="IDX161"></a>
</dt>
<dd><p><code>operatorp (<var>expr</var>, <var>op</var>)</code> returns <code>true</code>
if <var>op</var> is equal to the operator of <var>expr</var>.
</p>
<p><code>operatorp (<var>expr</var>, [<var>op_1</var>, ..., <var>op_n</var>])</code> returns <code>true</code>
if some element <var>op_1</var>, ..., <var>op_n</var> is equal to the operator of <var>expr</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>optimize</b><i> (<var>expr</var>)</i>
<a name="IDX162"></a>
</dt>
<dd><p>Returns an expression that produces the same value and
side effects as <var>expr</var> but does so more efficiently by avoiding the
recomputation of common subexpressions.  <code>optimize</code> also has the side
effect of &quot;collapsing&quot; its argument so that all common subexpressions
are shared.
Do <code>example (optimize)</code> for examples.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>optimprefix</b>
<a name="IDX163"></a>
</dt>
<dd><p>Default value: <code>%</code>
</p>
<p><code>optimprefix</code> is the prefix used for generated symbols by
the <code>optimize</code> command.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ordergreat</b><i> (<var>v_1</var>, ..., <var>v_n</var>)</i>
<a name="IDX164"></a>
</dt>
<dd><p>Sets up aliases for the variables <var>v_1</var>, ..., <var>v_n</var>
such that <var>v_1</var> &gt; <var>v_2</var> &gt; ...  &gt; <var>v_n</var>,
and <var>v_n</var> &gt; any other variable not mentioned as an
argument.
</p>
<p>See also <code>orderless</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ordergreatp</b><i> (<var>expr_1</var>, <var>expr_2</var>)</i>
<a name="IDX165"></a>
</dt>
<dd><p>Returns <code>true</code> if <var>expr_2</var> precedes <var>expr_1</var> in the
ordering set up with the <code>ordergreat</code> function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>orderless</b><i> (<var>v_1</var>, ..., <var>v_n</var>)</i>
<a name="IDX166"></a>
</dt>
<dd><p>Sets up aliases for the variables <var>v_1</var>, ..., <var>v_n</var>
such that <var>v_1</var> &lt; <var>v_2</var> &lt; ...  &lt; <var>v_n</var>,
and <var>v_n</var> &lt; any other variable not mentioned as an
argument.
</p>
<p>Thus the complete ordering scale is: numerical constants &lt;
declared constants &lt; declared scalars &lt; first argument to <code>orderless</code> &lt;
...  &lt; last argument to <code>orderless</code> &lt; variables which begin with A &lt; ...
&lt; variables which begin with Z &lt; last argument to <code>ordergreat</code> &lt;
 ... &lt; first argument to <code>ordergreat</code> &lt; declared <code>mainvar</code>s.
</p>
<p>See also <code>ordergreat</code> and <code>mainvar</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>orderlessp</b><i> (<var>expr_1</var>, <var>expr_2</var>)</i>
<a name="IDX167"></a>
</dt>
<dd><p>Returns <code>true</code> if <var>expr_1</var> precedes <var>expr_2</var> in the
ordering set up by the <code>orderless</code> command.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>part</b><i> (<var>expr</var>, <var>n_1</var>, ..., <var>n_k</var>)</i>
<a name="IDX168"></a>
</dt>
<dd><p>Returns parts of the displayed form of <code>expr</code>. It
obtains the part of <code>expr</code> as specified by the indices <var>n_1</var>, ..., <var>n_k</var>.  First
part <var>n_1</var> of <code>expr</code> is obtained, then part <var>n_2</var> of that, etc.  The result is
part <var>n_k</var> of ... part <var>n_2</var> of part <var>n_1</var> of <code>expr</code>.
</p>
<p><code>part</code> can be used to obtain an element of a list, a row of a matrix, etc.
</p>
<p>If the last argument to a part function is a list of indices then
several subexpressions are picked out, each one corresponding to an
index of the list.  Thus <code>part (x + y + z, [1, 3])</code> is <code>z+x</code>.
</p>
<p><code>piece</code> holds the last expression selected when using the part
functions.  It is set during the execution of the function and thus
may be referred to in the function itself as shown below.
</p>
<p>If <code>partswitch</code> is set to <code>true</code> then <code>end</code> is returned when a
selected part of an expression doesn't exist, otherwise an error
message is given.
</p>

<p>Example: <code>part (z+2*y, 2, 1)</code> yields 2.
</p>
<p><code>example (part)</code> displays additional examples.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>partition</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX169"></a>
</dt>
<dd><p>Returns a list of two expressions.  They are (1)
the factors of <var>expr</var> (if it is a product), the terms of <var>expr</var> (if it is a
sum), or the list (if it is a list) which don't contain var and, (2)
the factors, terms, or list which do.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) partition (2*a*x*f(x), x);
(%o1)                     [2 a, x f(x)]
(%i2) partition (a+b, x);
(%o2)                      [b + a, 0]
(%i3) partition ([a, b, f(a), c], a); 
(%o3)                  [[b, c], [a, f(a)]]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>partswitch</b>
<a name="IDX170"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>partswitch</code> is <code>true</code>, <code>end</code> is returned
when a selected part of an expression doesn't exist, otherwise an
error message is given.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>pickapart</b><i> (<var>expr</var>, <var>n</var>)</i>
<a name="IDX171"></a>
</dt>
<dd><p>Assigns intermediate expression labels to subexpressions of
<var>expr</var> at depth <var>n</var>, an integer.
Subexpressions at greater or lesser depths are not assigned labels.
<code>pickapart</code> returns an expression in terms of intermediate expressions
equivalent to the original expression <var>expr</var>.
</p>
<p>See also <code>part</code>, <code>dpart</code>, <code>lpart</code>, <code>inpart</code>, and <code>reveal</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));
                                          2
                                     sin(x )   b + a
(%o1)       - log(sqrt(x + 1) + 1) + ------- + -----
                                        3        2
(%i2) pickapart (expr, 0);

                                          2
                                     sin(x )   b + a
(%t2)       - log(sqrt(x + 1) + 1) + ------- + -----
                                        3        2

(%o2)                          %t2
(%i3) pickapart (expr, 1);

(%t3)                - log(sqrt(x + 1) + 1)


                                  2
                             sin(x )
(%t4)                        -------
                                3


                              b + a
(%t5)                         -----
                                2

(%o5)                    %t5 + %t4 + %t3
(%i5) pickapart (expr, 2);

(%t6)                 log(sqrt(x + 1) + 1)


                                  2
(%t7)                        sin(x )


(%t8)                         b + a

                         %t8   %t7
(%o8)                    --- + --- - %t6
                          2     3
(%i8) pickapart (expr, 3);

(%t9)                    sqrt(x + 1) + 1


                                2
(%t10)                         x

                  b + a              sin(%t10)
(%o10)            ----- - log(%t9) + ---------
                    2                    3
(%i10) pickapart (expr, 4);

(%t11)                     sqrt(x + 1)

                      2
                 sin(x )   b + a
(%o11)           ------- + ----- - log(%t11 + 1)
                    3        2
(%i11) pickapart (expr, 5);

(%t12)                        x + 1

                   2
              sin(x )   b + a
(%o12)        ------- + ----- - log(sqrt(%t12) + 1)
                 3        2
(%i12) pickapart (expr, 6);
                  2
             sin(x )   b + a
(%o12)       ------- + ----- - log(sqrt(x + 1) + 1)
                3        2
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>System variable:</u> <b>piece</b>
<a name="IDX172"></a>
</dt>
<dd><p>Holds the last expression selected when using the <code>part</code>
functions.
It is set during the execution of the function and thus
may be referred to in the function itself.
</p>

</dd></dl>

<dl>
<dt><u>Function:</u> <b>polarform</b><i> (<var>expr</var>)</i>
<a name="IDX173"></a>
</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>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>powers</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX174"></a>
</dt>
<dd><p>Gives the powers of <var>x</var> occuring in <var>expr</var>.
</p>
<p><code>load (powers)</code> loads this function.
</p>

</dd></dl>

<dl>
<dt><u>Function:</u> <b>product</b><i> (<var>expr</var>, <var>i</var>, <var>i_0</var>, <var>i_1</var>)</i>
<a name="IDX175"></a>
</dt>
<dd><p>Represents a product of the values of <var>expr</var> as
the index <var>i</var> varies from <var>i_0</var> to <var>i_1</var>.
The noun form <code>'product</code> is displayed as an uppercase letter pi.
</p>
<p><code>product</code> evaluates <var>expr</var> and lower and upper limits <var>i_0</var> and <var>i_1</var>,
<code>product</code> quotes (does not evaluate) the index <var>i</var>.
</p>
<p>If the upper and lower limits differ by an integer,
<var>expr</var> is evaluated for each value of the index <var>i</var>,
and the result is an explicit product.
</p>
<p>Otherwise, the range of the index is indefinite.
Some rules are applied to simplify the product.
When the global variable <code>simpproduct</code> is <code>true</code>, additional rules are applied.
In some cases, simplification yields a result which is not a product;
otherwise, the result is a noun form <code>'product</code>.
</p>
<p>See also <code>nouns</code> and <code>evflag</code>.
</p>
<p>Examples:
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) product (x + i*(i+1)/2, i, 1, 4);
(%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
(%i2) product (i^2, i, 1, 7);
(%o2)                       25401600
(%i3) product (a[i], i, 1, 7);
(%o3)                 a  a  a  a  a  a  a
                       1  2  3  4  5  6  7
(%i4) product (a(i), i, 1, 7);
(%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
(%i5) product (a(i), i, 1, n);
                             n
                           /===\
                            ! !
(%o5)                       ! !  a(i)
                            ! !
                           i = 1
(%i6) product (k, k, 1, n);
                               n
                             /===\
                              ! !
(%o6)                         ! !  k
                              ! !
                             k = 1
(%i7) product (k, k, 1, n), simpproduct;
(%o7)                          n!
(%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                             n
                           /===\
                            ! !    1
(%o8)                       ! !  -----
                            ! !  k + 1
                           k = 1
(%i9) product (if k &lt;= 5 then a^k else b^k, k, 1, 10);
                              15  40
(%o9)                        a   b
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>realpart</b><i> (<var>expr</var>)</i>
<a name="IDX176"></a>
</dt>
<dd><p>Returns the real part of <var>expr</var>. <code>realpart</code> and <code>imagpart</code> will
work on expressions involving trigonometic and hyperbolic functions,
as well as square root, logarithm, and exponentiation.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>rectform</b><i> (<var>expr</var>)</i>
<a name="IDX177"></a>
</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>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>rembox</b><i> (<var>expr</var>, unlabelled)</i>
<a name="IDX178"></a>
</dt>
<dt><u>Function:</u> <b>rembox</b><i> (<var>expr</var>, <var>label</var>)</i>
<a name="IDX179"></a>
</dt>
<dt><u>Function:</u> <b>rembox</b><i> (<var>expr</var>)</i>
<a name="IDX180"></a>
</dt>
<dd><p>Removes boxes from <var>expr</var>.
</p>
<p><code>rembox (<var>expr</var>, unlabelled)</code> removes all unlabelled boxes from <var>expr</var>.
</p>
<p><code>rembox (<var>expr</var>, <var>label</var>)</code> removes only boxes bearing <var>label</var>.
</p>
<p><code>rembox (<var>expr</var>)</code> removes all boxes, labelled and unlabelled.
</p>
<p>Boxes are drawn by the <code>box</code>, <code>dpart</code>, and <code>lpart</code> functions.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x);
                                  a d - b c
(%o1)                sin(%pi x) + ---------
                                      2
                                     h
(%i2) dpart (dpart (expr, 1, 1), 2, 2);
                        &quot;&quot;&quot;&quot;&quot;&quot;&quot;    a d - b c
(%o2)               sin(&quot;%pi x&quot;) + ---------
                        &quot;&quot;&quot;&quot;&quot;&quot;&quot;      &quot;&quot;&quot;&quot;
                                     &quot; 2&quot;
                                     &quot;h &quot;
                                     &quot;&quot;&quot;&quot;
(%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);
                  FOO&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;   BAR&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                  &quot;    &quot;&quot;&quot;&quot;&quot;&quot;&quot; &quot;   &quot;a d - b c&quot;
(%o3)             &quot;sin(&quot;%pi x&quot;)&quot; + &quot;---------&quot;
                  &quot;    &quot;&quot;&quot;&quot;&quot;&quot;&quot; &quot;   &quot;  &quot;&quot;&quot;&quot;   &quot;
                  &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;   &quot;  &quot; 2&quot;   &quot;
                                   &quot;  &quot;h &quot;   &quot;
                                   &quot;  &quot;&quot;&quot;&quot;   &quot;
                                   &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
(%i4) rembox (expr2, unlabelled);
                                  BAR&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                   FOO&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;   &quot;a d - b c&quot;
(%o4)              &quot;sin(%pi x)&quot; + &quot;---------&quot;
                   &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;   &quot;    2    &quot;
                                  &quot;   h     &quot;
                                  &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
(%i5) rembox (expr2, FOO);
                                  BAR&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                       &quot;&quot;&quot;&quot;&quot;&quot;&quot;    &quot;a d - b c&quot;
(%o5)              sin(&quot;%pi x&quot;) + &quot;---------&quot;
                       &quot;&quot;&quot;&quot;&quot;&quot;&quot;    &quot;  &quot;&quot;&quot;&quot;   &quot;
                                  &quot;  &quot; 2&quot;   &quot;
                                  &quot;  &quot;h &quot;   &quot;
                                  &quot;  &quot;&quot;&quot;&quot;   &quot;
                                  &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
(%i6) rembox (expr2, BAR);
                   FOO&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;
                   &quot;    &quot;&quot;&quot;&quot;&quot;&quot;&quot; &quot;   a d - b c
(%o6)              &quot;sin(&quot;%pi x&quot;)&quot; + ---------
                   &quot;    &quot;&quot;&quot;&quot;&quot;&quot;&quot; &quot;     &quot;&quot;&quot;&quot;
                   &quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;&quot;     &quot; 2&quot;
                                      &quot;h &quot;
                                      &quot;&quot;&quot;&quot;
(%i7) rembox (expr2);
                                  a d - b c
(%o7)                sin(%pi x) + ---------
                                      2
                                     h
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>sum</b><i> (<var>expr</var>, <var>i</var>, <var>i_0</var>, <var>i_1</var>)</i>
<a name="IDX181"></a>
</dt>
<dd><p>Represents a summation of the values of <var>expr</var> as
the index <var>i</var> varies from <var>i_0</var> to <var>i_1</var>.
The noun form <code>'sum</code> is displayed as an uppercase letter sigma.
</p>
<p><code>sum</code> evaluates its summand <var>expr</var> and lower and upper limits <var>i_0</var> and <var>i_1</var>,
<code>sum</code> quotes (does not evaluate) the index <var>i</var>.
</p>
<p>If the upper and lower limits differ by an integer,
the summand <var>expr</var> is evaluated for each value of the summation index <var>i</var>,
and the result is an explicit sum.
</p>
<p>Otherwise, the range of the index is indefinite.
Some rules are applied to simplify the summation.
When the global variable <code>simpsum</code> is <code>true</code>, additional rules are applied.
In some cases, simplification yields a result which is not a summation;
otherwise, the result is a noun form <code>'sum</code>.
</p>
<p>When the <code>evflag</code> (evaluation flag) <code>cauchysum</code> is <code>true</code>,
a product of summations is expressed as a Cauchy product,
in which the index of the inner summation is a function of the
index of the outer one, rather than varying independently.
</p>
<p>The global variable <code>genindex</code> is the alphabetic prefix used to generate the next index of summation,
when an automatically generated index is needed.
</p>
<p><code>gensumnum</code> is the numeric suffix used to generate the next index of summation,
when an automatically generated index is needed.
When <code>gensumnum</code> is <code>false</code>, an automatically-generated index is only
<code>genindex</code> with no numeric suffix.
</p>
<p>See also <code>sumcontract</code>, <code>intosum</code>,
<code>bashindices</code>, <code>niceindices</code>,
<code>nouns</code>, <code>evflag</code>, and <code>zeilberger</code>.
</p>
<p>Examples:
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) sum (i^2, i, 1, 7);
(%o1)                          140
(%i2) sum (a[i], i, 1, 7);
(%o2)           a  + a  + a  + a  + a  + a  + a
                 7    6    5    4    3    2    1
(%i3) sum (a(i), i, 1, 7);
(%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
(%i4) sum (a(i), i, 1, n);
                            n
                           ====
                           \
(%o4)                       &gt;    a(i)
                           /
                           ====
                           i = 1
(%i5) sum (2^i + i^2, i, 0, n);
                          n
                         ====
                         \       i    2
(%o5)                     &gt;    (2  + i )
                         /
                         ====
                         i = 0
(%i6) sum (2^i + i^2, i, 0, n), simpsum;
                              3      2
                   n + 1   2 n  + 3 n  + n
(%o6)             2      + --------------- - 1
                                  6
(%i7) sum (1/3^i, i, 1, inf);
                            inf
                            ====
                            \     1
(%o7)                        &gt;    --
                            /      i
                            ====  3
                            i = 1
(%i8) sum (1/3^i, i, 1, inf), simpsum;
                                1
(%o8)                           -
                                2
(%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                              inf
                              ====
                              \     1
(%o9)                      30  &gt;    --
                              /      2
                              ====  i
                              i = 1
(%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                  2
(%o10)                       5 %pi
(%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                            n
                           ====
                           \       1
(%o11)                      &gt;    -----
                           /     k + 1
                           ====
                           k = 1
(%i12) sum (if k &lt;= 5 then a^k else b^k, k, 1, 10));
Incorrect syntax: Too many )'s
else b^k, k, 1, 10))
                  ^
(%i12) linenum:11;
(%o11)                         11
(%i12) sum (integrate (x^k, x, 0, 1), k, 1, n);
                            n
                           ====
                           \       1
(%o12)                      &gt;    -----
                           /     k + 1
                           ====
                           k = 1
(%i13) sum (if k &lt;= 5 then a^k else b^k, k, 1, 10);
          10    9    8    7    6    5    4    3    2
(%o13)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>lsum</b><i> (<var>expr</var>, <var>x</var>, <var>L</var>)</i>
<a name="IDX182"></a>
</dt>
<dd><p>Represents the sum of <var>expr</var> for each element <var>x</var> in <var>L</var>.
</p>
<p>A noun form <code>'lsum</code> is returned
if the argument <var>L</var> does not evaluate to a list.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) lsum (x^i, i, [1, 2, 7]);
                            7    2
(%o1)                      x  + x  + x
(%i2) lsum (i^2, i, rootsof (x^3 - 1));
                     ====
                     \      2
(%o2)                 &gt;    i
                     /
                     ====
                                   3
                     i in rootsof(x  - 1)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>verbify</b><i> (<var>f</var>)</i>
<a name="IDX183"></a>
</dt>
<dd><p>Returns the verb form of the function name <var>f</var>.
</p>
<p>See also <code>verb</code>, <code>noun</code>, and <code>nounify</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) verbify ('foo);
(%o1)                          foo
(%i2) :lisp $%
$FOO
(%i2) nounify (foo);
(%o2)                          foo
(%i3) :lisp $%
%FOO
</pre></td></tr></table>
</dd></dl>

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