Intro 11. Power and Inverse

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11. Power and Inverse
The power conjunction`` (`^:`)`` is analogous to the power function`` (`^`). For example:

       ]a=: 10^ b=: i.5
    1 10 100 1000 10000

       b
    0 1 2 3 4

       %:a
    1 3.16228 10 31.6228 100

       %: %: a
    1 1.77828 3.16228 5.62341 10

       %: ^: 2 a
    1 1.77828 3.16228 5.62341 10

       %: ^: 3 a
    1 1.33352 1.77828 2.37137 3.16228

       %: ^: b a
    1      10     100    1000   10000
    1 3.16228      10 31.6228     100
    1 1.77828 3.16228 5.62341      10
    1 1.33352 1.77828 2.37137 3.16228
    1 1.15478 1.33352 1.53993 1.77828

       (cos=: 2&o.) ^: b d=:1
    1 0.540302 0.857553 0.65429 0.79348

       ] y=: cos ^: _ d
    0.739085

       y=cos y
    1

Successive applications of`cos` appear to be converging to a limiting value; the infinite power`` (`cos ^: _`)`` yields this limit.
A right argument of`_1` produces the inverse function. Thus:

       %: ^: _1 b
    0 1 4 9 16

       *: b
    0 1 4 9 16

       %: ^: (-b) b
    0 1     2         3         4
    0 1     4         9        16
    0 1    16        81       256
    0 1   256      6561     65536
    0 1 65536 4.30467e7 4.29497e9

Exercises

11.1 | The square function`*:` is the inverse of the square root function`%:` and`%:^:_1` is therefore equivalent to`*:` . Look for, and experiment with, other inverse pairs among the primitive functions in the Vocabulary.

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