Vocabulary/hat

`^ y`	Exponential

Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

e^y, the y-th power of the mathematical constant: e .

The antilogarithm of a natural logarithm.

   ^1
2.71828
   ^2
7.38906
   ^ ^. 123.45
123.45

Combinations using ^ y that have exceptionally good performance include:

What it does	Type; Precisions; Ranks	Syntax	Variants; Restrictions	Benefits; Bug Warnings
e^πy		`^@o. y`		handles large values of `y`

`x ^ y`	Power

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

x^y, the y-th power of numeric noun: x .

   2^8
256

Logarithm (x ^. y)

1. Some results are defined in J that are often left undefined:

2. Taking a rational power of an extended integer produces a floating-point result whenever the denominator of the power is not 1.

Combinations using x ^ y that have exceptionally good performance include:

What it does	Type; Precisions; Ranks	Syntax	Variants; Restrictions	Benefits; Bug Warnings
Integer powers	non-complex `x`, integer `y`	`x ^ y`		Uses repeated multiplication (avoids log)
Powers mod(m)	integer, extended integer	`x m&\|@^ y` `m&\|@(n&^) y`		Avoids the large result of exponentiation

`x ^!.p y`	Stope Function

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

x ^!.p y is x*(x+p)*(x+2*p)*... for y terms.

   5 ^!.1 (3)   NB. 5 * 6 * 7
210

1. Calculate the number of permutations of x things taken y at a time, _xP_y

   5 ^!._1 (3)   NB. 5 * 4 * 3
60

2. Calculate the rising Pochhammer symbol (x)_y as used in the hypergeometric function

   5.1 ^!.1 (3)  NB. 5.1 * 6.1 * 7.1
220.881

1. y must be nonnegative and tolerantly equal to an integer.

2. x ^!.p y is the same as */x + p*i. y .

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