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).

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