Is {e^x} periodic function?
no. Because , if it had a period, say k > 0, then that would mean that
e^(x+k) = e^x
e^x e^k = e^x
e^k = 1
k=0
But, we specified k > 0.
I mean, just look at the graph. Surely it is not periodic!
Now, if we use a complex number z, then it is true that e^z is periodic.
To determine if the function f(x) = e^x is periodic, we need to understand the definition of a periodic function.
A periodic function is one that repeats its values at regular intervals. Mathematically, a function f(x) is periodic if there exists a positive number P such that f(x) = f(x + P) for all x in the domain of f(x).
Now, let's consider the function f(x) = e^x.
To show that f(x) = e^x is not periodic, we can prove that there is no positive number P for which f(x) = f(x + P) for all x in the domain.
Taking the derivative of f(x), we have f'(x) = e^x.
Since the derivative f'(x) is equal to the original function f(x), this means that the function f(x) = e^x does not repeat its values at regular intervals.
Therefore, we can conclude that f(x) = e^x is not a periodic function.