# A bit more on ‘a power b’

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Just to take off from the previous post, when $a, b$ are complex numbers, a question was raised as to whether it makes sense to define $a^b$.

It can be defined and the result is a well defined complex number. I had given the value for $i^i$. A generalised result can be derived as follows. I will use $z_1, z_2$ instead of $a, b$.

Let $z_1 = a + i b = r_1 e^{i \theta_1}$

and $z_2 = c + i d = r_2 e^{i \theta_2}$

Then, $z_1^{z_2} = {(a+ib)}^{(c+id)}$

$= {(r_1 e^{i \theta_1})}^{c+id}$

$= r_1^c r_1^{id} e^{i \theta_1 (c + id)}$

$= r_1^c e^{\ln r_1 id} e^{-d \theta_1 + i c \theta_1}$

$= (r_1^c e^{-d \theta_1}) e^{i(c \theta_1 +d \ln r_1)}$

You can get $i^i$ pop out of the above if you substitute appropriate values $(r_1 = 1, \theta_1 = \pi/2, c = 0, d = 1)$.