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Power u^:n  _ _ _  

Two cases occur: a numeric integer n, and a gerund n .
 

Numeric case. The verb u (or x&u) is applied n times. For example:
   (] ; +/\ ; +/\^: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|
+---------+-----------+------------+-------------+
An infinite power n produces the limit of the application of u . For example, if x=:2 and y=:1, then x o.^:_ y is 0.73908, the solution of the equation y=Cos y . If n is negative, the obverse u^:_1 is applied |n times. The obverse (which is normally the inverse) is specified for six cases:

1.  The self-inverse functions + - -. % %. |. |: /: [ ] C. p.

2.  The pairs in the following tables:
 
<   >
<:   >:
+.   j./"1"_
+:   -:
*.   r./"1"_
*:   %:
^   ^.
$.   $.^:_1
,:   {.
;:   ;@(,&' '&.>"1)
#.   #:
 
!   3 : '(-(!-y."_)%1e_3&* !"0 D:1 ])^:_^.y.'
3!:1   3!:2
3!:3   3!:2
\:   /:@|.
".   ":
j.   %&0j1
o.   %&1p1
p:   π(n)
q:   */
r.   %&0j1@^.
s:   5&s:
u:   3&s:
x:   _1&x:
 
+~   -:
*~   %:
^~   3 : '(- -&b@(*^.) % >:@^.)^:_ b=.^.y.'"0
,~   <.@-:@# {. ]
,:~   {.
;~   >@{.
j.~   %&1j1

3.  Obviously invertible bonded dyads such as -&3 and 10&^. and 1 0 2&|: and 3&|. and 1&o. and a.&i. as well as u@v and u&v if u and v are invertible.

4.  Monads of the form v/\ and v/\. where v is one of + * - % = ~:

5.  Obverses specified by the conjunction :.

6.  The following cases merit special mention:

  p:^:_1 n gives the number of primes less than n, denoted by π(n) in math
 
  q:^:_1 is */
 
  b&#^:_1 where b is a boolean list is "Expand" (whose fill atom f can be specified by fit, b&#^:_1!.f)
 
  a&#.^:_1 produces the base-a representation
 
  !^:_1 and !&n^:_1 and !&n&^:_1 produce the appropriate results
 

Gerund case. (Compare with the gerund case of the merge adverb })

   x u^:(v0`v1`v2)y      (x v0 y)u^:(x v1 y) (x v2 y)
   x u^:(   v1`v2)y      x u^:([` v1`v2) y
     u^:(   v1`v2)y      u^:(v1 y) (v2 y)



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