What is the geometrical meaning of $\ i^i $? What is the geometrical meaning of $ i^i $ ? Is there any meaning at all? 
If I write $ i= e^{i* \pi/2}$, I get $ i^i =  e^{- \pi/2}$.
But if I write $ i= e^{5i \pi/2}$ I get its value  $e^{-5 \pi/2}$.
Since $i^i$ cannot be defined uniquely, it appears to me that it should not be defined. Also, what geometrical operation is going on here?
In case this is defined, can you show me any of its applications?
 A: $i^i$ is multivalued !
First, we need the logarithms of $i$. They are given by
$\log i =\log|i|+i \arg i+2k \pi i=\frac{i \pi}{2}+2 k  \pi i$ with $k \in \mathbb Z.$
Then we have 
$i \log i=\frac{- \pi}{2}-2 k  \pi $ with $k \in \mathbb Z.$
Hence
$$i^i \in \{e^{- \pi/2}e^{-2k \pi}: k \in \mathbb Z\}.$$
A: The problem comes down to how you are to define $u^w$ when $u$ and $w$ both are nonreal complex numbers. The only way would be $\exp(w\log(u))$, where I’m writing $\exp(z)$ for $e^z$. No problem ever in defining $\exp(z)$, it’s a good everywhere-defined function. But that can not be said for the logarithm, as you probably know. We do use the log, with due caution because it has no consistent definition for all nonzero complex numbers. That means, for $\log(z)$, choosing one of the infinitely many numbers $w$ such that $\exp(w)=z$.
Same thing for $u^w$: it is not well defined, but as long as we’re aware that there are infinitely many possible values, we may be able to use the expression fruitfully. But always with care.
A: Let us consider the expression
$$z^i$$ for some complex $z$, and let $z=re^{i\theta}$.
Using the logarithm, we have
$$z^i=e^{i(\log r+i\theta)}=e^{-\theta+i\log r}.$$
This is a complex number, let $se^{i\phi}$ with 
$$s=e^{-\theta},\\e^{\phi}=r.$$
These two relations describe logarithmic spirals, with opposite orientation.
To construct $z^i$,


*

*draw the logarithmic spiral (blue);

*draw the vector $z^*$ (conjugate, green);

*draw the line of support of $z^*$ (green) and intersect the spiral, to get the modulus;

*draw a circle centered at the origin, through $z^*$ (green), and intersect the spiral, to get the argument;

*the requested $z^i$ is the intersection (red) of the circle by the first point (orange) and the line by the second (orange).

