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.

For non-integer exponents, exponentiation $x^y$ is only well defined (denotes a single value) when $x$ is a positive real number; on the other hand $y$ can be any complex number. (And for most purposes one can do with just using $x=e=\exp(1)$, but that is a separate matter). Assuming that, the rule $x^{y+z}=x^yx^z$ is always valid. However the rule for multiplication in the exponent has a restriction that is rarely stated explicitly:
$$
  x^{yz}=(x^y)^z \qquad
  \text{provided that $x>0$ is real} \textbf{ and that } \text{$y$ is real.}
$$
Note that it does not suffice that the expressions on both sides are defined, in other words that $x$ and $x^y$ are positive real numbers: both sides can be well defined value, yet different. The example in the question illustrates this (for any non-integer value of$~k$).
See this answer to a similar question for more details, and a proof of the rule stated above.
