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 Exponential ^  0 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 phrasex ^ 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 5 of [14] 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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