fractional iterates of arg(z) and |z|
#1
Is it possible to define a function
h(h(z)) = |z|, h(z) =/= |z|

or
h(h(z)) = arg(z)

I wonder because these seem like difficult functions to crack, considering they have no Taylor series.
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#2
(03/31/2011, 08:28 PM)JmsNxn Wrote: Is it possible to define a function
h(h(z)) = |z|, h(z) =/= |z|

or
h(h(z)) = arg(z)

I wonder because these seem like difficult functions to crack, considering they have no Taylor series.

We could set h(z) = |z| for z with non-negative real part, h(z) = -iz for z with negative real and non-negative imaginary part, and h(z) = iz for z with negative and imaginary parts. Then we'd get h(h(z)) = |z| for all z. A similar technique should work for h(h(z)) = arg(z).
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#3
Tongue that was easy
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