Evaluate the complex number into real and imaginary parts

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 $=e^{-\frac{\pi}{4\sqrt{2}}}\sin\frac{\pi}{4\sqrt{2}}$