Conjugate of a complex number

Question

Can you guys please tell that what's the conjugate of $i^i$? I tried to solve on the basis that multiplying conjugate with it's respective complex no. yields real no. So my try gives its answer as $i^{-i}$. Is it correct?

Answer 1

Short, hand-wavy answer

$$ \overline{i^i}={\overline i}^{\overline i}=(1/i)^{-i}=i^i $$ so the conjugate of $i^i$ is itself. I should mention that I got this trick from a different answer on this site, but I do not remember which one.

Rigorous answer

The definition of $z^w$ is $\exp(w\log z)=\exp(w(\log |z|+i\arg(z))$. This is a multivalued funciton, because $\text{arg}$ is multivalued, since it is only well defined up to a multiple of $2\pi$. Therefore, $$ i^i=\exp(i(\log |i|+i\arg i))=\exp(-(\pi/2+2\pi k)), $$ where the above equation is valid for any $k\in \mathbb Z$. No matter which $k$ you choose, the result is real, so the conjugate of $i^i$ is always itself.

Answer 2

We can compute $i^i=e^{\frac{-\pi}{2}+2k\pi}, k \in \Bbb Z$ .

What is worth noting is that, despite it being a highly irrational number, all values of $i^i$ are real - they don't have a complex conjugate, or at the very least, its conjugate is itself (I don't know for certain which is the better definition).