I am working on a complex number problem and need some guidance on how to calculate the principal value of the expression "$i^{\log(-i)}$". I understand that the principal value of a complex logarithm is the one with an argument in the range $(-π, π]$, but I'm not sure how to apply this concept to the given expression.
To be more specific, I'm looking for a step-by-step explanation or a mathematical approach to find the principal value of this complex exponentiation. Any insights, suggestions, or formulas that can help me compute this value would be greatly appreciated.
Thank you in advance for your assistance!
Let's start by computing the principal branch of $\log(-i)$: $$ \log(-i)=\log(e^{-i\pi/2})=-i\frac{\pi}{2}. \tag{1} $$ Now we do the same for $i^{\log(-i)}$: $$ i^{\log(-i)}=i^{-i\pi/2}=e^{-i(\pi/2)\log(i)}=e^{-i(\pi/2)\log(e^{i\pi/2})}= e^{-i(\pi/2)(i\pi/2)}=e^{\pi^2/4}. \tag{2} $$