download problem 2
Problem 1:
$i^{i}=exp(iln(i))=e^{-\pi/2}$
and
$1^{i}=exp(iln(1))=e^{0}=1$
and
$i^{0}=exp(0ln(i))=e^{0}=1$
and
$0^{i}=exp(iln(0))=e^{-i\infty}$
$e^{-i\infty}=Cos(\infty)-iSin(\infty)$
Problem 3:
$f(z)=z^{i} , f^{2}(z)=f(f(z))$
$f^{n}(i)=?$
$f^{1}(i)=i^{i}=e^{-\pi/2}$
$f^{2}(i)=(e^{-\pi/2})^{i}=e^{-i\pi/2}=-i$
$f^{3}(i)=-i^{i}=e^{\pi/2}$
$f^{4}(i)=(e^{\pi/2})^{i}=e^{i\pi/2}=i$
$f^{5}(i)=i^{i}=e^{-\pi/2}$
$f^{6}(i)=(e^{-\pi/2})^{i}=e^{-i\pi/2}=-i$
...
So:
$f^{1+4n}(i)=f^{1}(i)=e^{-\pi/2}$
$f^{2+4n}(i)=f^{2}(i)=-i$
$f^{3+4n}(i)=f^{3}(i)=e^{\pi/2}$
$f^{4+4n}(i)=f^{4}(i)=i$
