Euler's formula, is this true? *I've changed this question as below.
Let me have a function such as $ f(k) = \exp(j 2 \pi k ) $,  where $k$ is real value.
Using Euler's formula, we can write $f(k)$ as below,
$$ f(k) = \exp(j  2 \pi  k ) = \cos(2\pi k)+j \sin(2\pi k).$$
If $k$ is integer, this always goes to 1.
Up to here, nothing is weird and makes sense.
However, if the equation goes to
$$f(k) = \exp(j2\pi k)=e^{j2\pi k}=(e^{j2\pi})^k = (\cos(2\pi)+j\sin(2\pi))^k=1^k=1$$
What I want to know is this above equation makes sense or not.
Thank you in advance.

No, in complex numbers you may not assume that $(e^a)^b=e^{(ab)}$. You just found a counter-example.
