^ Exponential - Power

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Exponential | ^ _ 0 0 | Power

`^y` is equivalent to`e^y`,`` where `e` is Euler’s number`^1` (approximately 2.71828). The natural logarithm`` (`^.`)`` is inverse to`^` (that is,`y=^.^y` and `y=^^.y`). The monad`x&^` is inverse to the monad `x&^.` .`` For example:
       10&^ 10&^. 1 2 3 4 5
    1 2 3 4 5

       10&^. 10&^ 1 2 3 4 5
    1 2 3 4 5
|  | `x^2` and`x^3` and`x^0.5` are the square, cube, and square root of`x` . In general,`x^y` is`^y*^.x`,`` applying for complex numbers as well as real. For a non-negative integer`y`,`` the phrase`x ^ y` is equivalent to`*/y # x`;`` in particular,`*/` on an empty list is`1`,`` and`x^0` is`1` for any`x`,`` including`0` . The fit conjunction applies to`^` to yield a stope defined as follows:`x!.k n` is`*/x + k*i. n` .`` In particular,`^!._1` is the falling factorial function.

The last result in the first example below illustrates the falling factorial function, formed by the fit conjunction. See Chapter 4 of [8] for the use of stope functions, stope polynomials, and Stirling numbers in the difference calculus:

       e=: ^ 1 [ x=: 4 [ y=: 0 1 2 3
       ,.&.> x (e"_ ; e&^@] ; ^ ; ^@(] * ^.@]) ; (]^]) ; ^!._1) y
    +-------+-------+--+--+--+--+
    |2.71828|      1| 1| _| 1| 1|
    |       |2.71828| 4| 1| 1| 4|
    |       |7.38906|16| 4| 4|12|
    |       |20.0855|64|27|27|24|
    +-------+-------+--+--+--+--+

       S2=: %.@S1=: (^!._1/~ %. ^/~) @ i. @ x:
       (S1;S2) 8
    +---------------------------+-------------------+
    |1 0  0  0  0   0    0     0|1 0 0 0 0  0  0   0|
    |0 1 _1  2 _6  24 _120   720|0 1 1 1 1  1  1   1|
    |0 0  1 _3 11 _50  274 _1764|0 0 1 3 7 15 31  63|
    |0 0  0  1 _6  35 _225  1624|0 0 0 1 6 25 90 301|
    |0 0  0  0  1 _10   85  _735|0 0 0 0 1 10 65 350|
    |0 0  0  0  0   1  _15   175|0 0 0 0 0  1 15 140|
    |0 0  0  0  0   0    1   _21|0 0 0 0 0  0  1  21|
    |0 0  0  0  0   0    0     1|0 0 0 0 0  0  0   1|
    +---------------------------+-------------------+

`S1` gives (signed) Stirling numbers of the first kind and`S2` gives Stirling numbers of the second kind. They can be used to transform between ordinary and stope polynomials. Note that`x:` gives extended precision.

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