| 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. |
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.