So I've been trying to find the definition of function of exponential order and I found it in various places –> and some variations in said places (???). So a definition I found says:
$\left| {f\left( t \right)} \right| \le C{e^{at}},\;{\forall _a} \in \mathbb{R},\;C > 0,\;t > T > 0$
[ This would mean that $a$ can be any Real number, is C some fixed calculated Real number $> 0$, and this would work as of some calculated Real $t > 0$ (which would be the $T$ constant). Is this interpretation correct? ]
Other places say $a$ is some positive constant, others that this must hold for $t \ge 0$ instead of just $>$ (I assume that means ${\forall _t} \in {\mathbb{R_0}^ + }$, right?), others where no constant $T$ is mentioned and others that the first $\le$ sign is actually just $<$.
So please, can anyone tell me the actual definition of this? Some trusted book where it is or something? Or it doesn't really matter these things I said? (for some reason which I'd like to know too)
Thanks in advance for any help!
Best Answer
First of all, the formula seems strange as $T$ is unused and there should probably be $f(a)$ on the left.
But to the question itself: not sure if there is any standard definition. Exponential function usually refers to $x\mapsto \exp(x)$, or $x\mapsto \exp(tx)$ for some constant $t$.
So probably you can say that $f(x)$ grows no more then exponentially, if, for some $C$ and some $t>0$, we would have $$ f(x) \leq C \exp(tx). $$ This is similar to the big O notation in computer science -- it doesn't indicate that the function grows exponentially, it only indicates an upper bound.
Maybe for "exponential order", some people would understand that the function grows really exponentially and not less. This is formalized in the Big Theta notation here.