Evaluate the complex number into real and imaginary parts $\sqrt{i}^\sqrt{i}$. 

Solution

From indices we know that $\sqrt{i}=i^\frac{1}{2}$
$i^\frac{1}{2}=[\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}]^\frac{1}{2}=\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}$.
 


Therefore, 



$=e^{-\frac{\pi}{4\sqrt{2}}}.e^{i\frac{\pi}{4\sqrt{2}}}=e^{-{\frac{\pi}{4\sqrt{2}}}}.[\cos\frac{\pi}{4\sqrt{2}}+i\sin\frac{\pi}{4\sqrt{2}}]$
Therefore, the
Real part $=e^{-\frac{\pi}{4\sqrt{2}}}\cos\frac{\pi}{4\sqrt{2}}$
Imaginary part $=e^{-\frac{\pi}{4\sqrt{2}}}\sin\frac{\pi}{4\sqrt{2}}$