what will be the decomposition of the function $f(x) = i^x$

Question

I am just started to learn Complex Analysis.

We know that every complex valued functions whose domain is also complex number can be decomposed as $f(x)=f_1(x) + i f_2(x)$. Where $f_1(x)$ and $f_2(x)$ are real valued.

Then my question is what will be the decomposition of the function $f(x) = i^x$ ?

Please make some edit for me.

Answer 1

Since $$i=e^{i\pi /2} = \cos ( \pi/2)+ i \sin (\pi/2) $$, we have $$i^x=e^{i\pi x /2 } =\cos(\pi x/2 ) + i \sin ( \pi x/2 )$$

Thus $f_1(x) = \cos(\pi x/2 )$ and $f_2(x) = \sin(\pi x/2 )$