Question

why complex exponential function is periodic while real exponential function is non periodic ??

Answer 1

In polar co-ordinates, e^z is
[e^r] times\[e^i\theta\]
The imaginary part is periodic, because it is an expression involving sins and cosines. The real part does not involve sins and cosines.

Another answer: The series expansion for e^x for x real involves only positive terms, all of them powers of x, and so the bigger the value of x, the bigger e^x.

The series for x imaginary has i^2 recurring in it, so that it has a mixture of positive and negative terms. These combine to make the functiion periodic, exactly as is the case with sin and cos. [because it is the series expansion for sin and cosine, as you know].