Complex numbers and exponites

[Rekna]

If Z is a complex number how would I calculate:
Z^Z.

[Yakk]

Originally Posted by Rekna

If Z is a complex number how would I calculate:
Z^Z.

http://mathworld.wolfram.com/ComplexExponentiation.html

If I remember correctly there is a family of solutions.

Do you know how to raise a complex number to a real power?
Do you know how to raise a real number to a complex power?

[Chamaeleontidae]

This might be helpful to you... http://www.math.toronto.edu/mathnet/...omplexexp.html

[Yakk]

i = e^(pi*i*(1/2 + 2*k))

i^(a+b*i)
= e^(pi*i*(1/2 + 2*k))^(a+b*i)
= e^(pi*(1/2 + 2k)*(-b+a*i))
= e^(-pi*b*(1/2+2k) + pi*i*a*(1/2+2k))
= e^(-pi*b*(1/2+2k)) * e^(pi*i*a*(1/2+2k))

If b is not 0, this is a set of points arranged in a spiral. The magnatude of each point is
e^(-pi*b*(1/2+2k))
while the angle is
e^(pi*i*a*(1/2+2k))

If a is rational, the result takes on finitely many angles. If a is irrational, the angles form a dense net. Ie, if b is 0 and a is irrational, then the set of roots is dense on the complex unit circle.

If b is not zero, then the set of roots spirals from 0 out to infinity.

This seems less than useful.

You could identify a particular root as the primary root.