Euler's formula and $i^x = \cos(x \cdot \frac{\pi}{2})$ While playing around with a plotting software, i just found out that 
$$f(x) = i^x = \cos(x·\frac{\pi}{2})$$


*

*How does this connect to Euler's formula?

*Obviously, here, the alternating sign change is responsible for periodicity and form of the cosine. Is this also true for Euler's formula?


Please don't beat me, i'm an engineering student.
 A: The quantity $i^x$ by itself is not well-defined. The way one would like to define it is $i^x = e^{x\log i}$, and then use the Taylor series for the exponential to compute $e^{x\log i}$. The problem with this is that $\log i$ is not well-defined: there are infinitely many possible values of $\log i$, namely $$\log i = \frac{\pi i}{2} + 2\pi in$$ for any $n\in \mathbb{Z}$. Thus to define $i^x$, you have to make a choice as to which one of these logarithms you are using. The standard choice would be $\log i = \pi i/2$. In this case, $$i^x = e^{x\log i} = e^{i\pi x/2} = \cos(\pi x/2) + i\sin(\pi x/2).$$ However, if you had chosen $\log i = \pi i/2 + 2\pi in$ for some $n\neq 0$, then $$i^x = e^{x(\pi i/2 + 2\pi in)} = \cos(\pi x/2 + 2\pi nx) + i\sin(\pi x/2 + 2\pi nx).$$ 
A: My answer is not about your questions, which are wonderfully addressed in the answer and comments before me, but about your discovery. This should be a good warning: a lot of computer systems plot only the real part of a given expression. For example, it is true for Maple. Therefore, students should be careful because sometimes what is shown in the plot is not exactly in the original expression.
