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<br>
<table width=100%><tr>
<td align=left   width=33%><b><font size=+2>Power</font></b></td>
<td align=center width=34%><font face="Courier New" size=+3>u^:n &nbsp;_ _ _</font></td>
<td align=right  width=33%>&nbsp;</td>
</tr></table>
<br>

<table width=100%>
<tr><td valign=top bgcolor=#eeeeee>
Two cases occur: a numeric integer<tt> n</tt>,<tt> </tt>and a 
gerund<tt> n</tt> .
<br>&nbsp;</td>
</tr></table><br>

<b>Numeric case. </b>The verb<tt> u </tt> (or<tt> x&u</tt>)<tt> </tt>is 
applied<tt> n </tt>times.  For example:<br>

<pre>
   (] ; +/\ ; +/\^:2 ; +/\^:0 1 2 3 _1 _2 _3 _4) 1 2 3 4 5
+---------+-----------+------------+-------------+
|1 2 3 4 5|1 3 6 10 15|1 4 10 20 35|1  2  3  4  5|
|         |           |            |1  3  6 10 15|
|         |           |            |1  4 10 20 35|
|         |           |            |1  5 15 35 70|
|         |           |            |1  1  1  1  1|
|         |           |            |1  0  0  0  0|
|         |           |            |1 _1  0  0  0|
|         |           |            |1 _2  1  0  0|
+---------+-----------+------------+-------------+
</pre>

An infinite power<tt> n </tt>produces the limit of the application 
of<tt> u</tt> .<tt> </tt>For example, if<tt> x=:2 </tt>
and<tt> y=:1</tt>,<tt> </tt>
then<tt> x o.^:_ y </tt>is<tt> 0.73908</tt>,<tt> </tt>the solution of 
the equation<tt> y=Cos y</tt> .<tt> </tt>If<tt> n </tt>is negative, 
the obverse<tt> u^:_1 </tt>is applied<tt> |n </tt>times.  
The obverse (which is normally the inverse) is specified for six cases:<br><br>

1. &nbsp;The self-inverse functions<tt> + - -. % %. |. |: /: [ ] C. p.</tt><br><br>


2. &nbsp;The pairs in the following tables:<br>

<table><tr>
<td>&nbsp;</td>
<td valign=top><table>
<tr><td><tt>&lt;</tt></td>  <td>&nbsp;</td> <td><tt>&gt;</tt></td></tr>
<tr><td><tt>&lt;:</tt></td> <td>&nbsp;</td> <td><tt>&gt;:</tt></td></tr>
<tr><td><tt>+.</tt></td>    <td>&nbsp;</td> <td><tt>j./"1"_</tt></td></tr>
<tr><td><tt>+:</tt></td>    <td>&nbsp;</td> <td><tt>-:</tt></td></tr>
<tr><td><tt>*.</tt></td>    <td>&nbsp;</td> <td><tt>r./"1"_</tt></td></tr>
<tr><td><tt>*:</tt></td>    <td>&nbsp;</td> <td><tt>%:</tt></td></tr>
<tr><td><tt>^</tt></td>     <td>&nbsp;</td> <td><tt>^.</tt></td></tr>
<tr><td><tt>$.</tt></td>    <td>&nbsp;</td> <td><tt>$.^:_1</tt></td></tr>
<tr><td><tt>,:</tt></td>    <td>&nbsp;</td> <td><tt>{.</tt></td></tr>
<tr><td><tt>;:</tt></td>    <td>&nbsp;</td> <td><tt>;@(,&' '&.&gt;"1)</tt></td></tr>
<tr><td><tt>#.</tt></td>    <td>&nbsp;</td> <td><tt>#:</tt></td></tr>
</table></td>
<td>&nbsp;</td>
<td valign=top><table>
<tr><td><tt>!</tt></td>     <td>&nbsp;</td> <td><tt>3 : '(-(!-y."_)%1e_3&* !"0 D:1 ])^:_^.y.'</tt></td></tr>
<tr><td><tt>3!:1</tt></td>  <td>&nbsp;</td> <td><tt>3!:2</tt></td></tr>
<tr><td><tt>3!:3</tt></td>  <td>&nbsp;</td> <td><tt>3!:2</tt></td></tr>
<tr><td><tt>\:</tt></td>    <td>&nbsp;</td> <td><tt>/:@|.</tt></td></tr>
<tr><td><tt>".</tt></td>    <td>&nbsp;</td> <td><tt>":</tt></td></tr>
<tr><td><tt>j.</tt></td>    <td>&nbsp;</td> <td><tt>%&0j1</tt></td></tr>
<tr><td><tt>o.</tt></td>    <td>&nbsp;</td> <td><tt>%&1p1</tt></td></tr>
<tr><td><tt>p:</tt></td>    <td>&nbsp;</td> <td>&pi;<tt>(n)</tt></td></tr>
<tr><td><tt>q:</tt></td>    <td>&nbsp;</td> <td><tt>*/</tt></td></tr>
<tr><td><tt>r.</tt></td>    <td>&nbsp;</td> <td><tt>%&0j1@^.</tt></td></tr>
<tr><td><tt>s:</tt></td>    <td>&nbsp;</td> <td><tt>5&s:</tt></td></tr>
<tr><td><tt>u:</tt></td>    <td>&nbsp;</td> <td><tt>3&u:</tt></td></tr>
<tr><td><tt>x:</tt></td>    <td>&nbsp;</td> <td><tt>_1&x:</tt></td></tr>
</table></td>
</tr></table>

<table><tr>
<td>&nbsp;</td>
<td><table>
<tr><td><tt>+~</tt></td>    <td>&nbsp;</td> <td><tt>-:</tt></td></tr>
<tr><td><tt>*~</tt></td>    <td>&nbsp;</td> <td><tt>%:</tt></td></tr>
<tr><td><tt>^~</tt></td>    <td>&nbsp;</td> <td><tt>3 : '(- -&b@(*^.) % >:@^.)^:_ b=.^.y.'"0</tt></td></tr>
<tr><td><tt>,~</tt></td>    <td>&nbsp;</td> <td><tt>&lt;.@-:@# {. ]</tt></td></tr>
<tr><td><tt>,:~</tt></td>   <td>&nbsp;</td> <td><tt>{.</tt></td></tr>
<tr><td><tt>;~</tt></td>    <td>&nbsp;</td> <td><tt>>@{.</tt></td></tr>
<tr><td><tt>j.~</tt></td>   <td>&nbsp;</td> <td><tt>%&1j1</tt></td></tr>
</table></td>
</tr></table>
<br>

3. &nbsp;Obviously invertible bonded dyads such 
as<tt> -&3 </tt>and<tt> 10&^. </tt>
and<tt> 1 0 2&|: </tt>and<tt> 3&|. </tt>and<tt> 1&o. </tt>
and<tt> a.&i. </tt>as well as<tt> u@v </tt>and<tt> u&v </tt>
if<tt> u </tt>and<tt> v </tt>are invertible.<br><br>

4. &nbsp;Monads of the form<tt> v/\ </tt>and<tt> v/\. </tt>where<tt> v </tt>
is one of<tt> +  * -  %  =  ~:</tt><br><br>

5. &nbsp;Obverses specified by the conjunction<tt> :.</tt><br><br>

6. &nbsp;The following cases merit special mention:<br><br>

<table>
<tr><td>&nbsp;</td><td valign=top>
<tt>p:^:_1 n </tt>gives the number of primes less than<tt> n</tt>,<tt> </tt>
denoted by<tt> </tt>&pi;<tt>(n) </tt>in math<br>&nbsp;</td></tr>

<tr><td>&nbsp;</td><td valign=top>
<tt>q:^:_1 </tt>is<tt> */</tt><br>&nbsp;</td></tr>

<tr><td>&nbsp;</td><td valign=top>
<tt>b&#^:_1 </tt>where<tt> b </tt>is a boolean list is "Expand" 
(whose fill atom<tt> f </tt>can be specified by 
<i>fit</i>,<tt> b&amp;#^:_1!.f</tt>)<br>&nbsp;</td></tr>

<tr><td>&nbsp;</td><td valign=top>
<tt>a&#.^:_1 </tt>produces the base-<tt>a </tt>representation<br>&nbsp;</td></tr>

<tr><td>&nbsp;</td><td valign=top>
<tt>!^:_1 </tt>and<tt> !&amp;n^:_1 </tt>and<tt> n&amp;!^:_1 </tt>
produce the appropriate results<br>&nbsp;</td></tr>
</table>
<br>

<b>Gerund case.</b>  (Compare with the gerund case of the merge 
adverb<tt> }</tt>)<br><br>

<pre>
   x u^:(v0`v1`v2)y   <font face=Times size=+1>&harr;</font>   (x v0 y)u^:(x v1 y) (x v2 y)
   x u^:(   v1`v2)y   <font face=Times size=+1>&harr;</font>   x u^:([` v1`v2) y
     u^:(   v1`v2)y   <font face=Times size=+1>&harr;</font>   u^:(v1 y) (v2 y)
</pre>


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