# Actual definition of function of exponential order?

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!

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.

• Thanks for the reply! Why do you say $f(a)$ on the left? Because the variable here should be t (actually I should have said I was talking in Time domain, not in Real domain, but I was messing with Laplace Transforms and forgot to say that, my bad). So in Real domain $x$ and $C$ can be any numbers from $\mathbb{R}$ and your $t$ (my $a$) has to be positive? (Because on my post, they say $a$ can be any number.) Sorry, this is confusing me a bit, even after reading about the Big-O notation. Btw on the Big-O notation, they say $x >= x_0$. That $x_0$ would be the $T$ I wrote up there, I think. Commented Oct 18, 2020 at 19:32
• So if it is $t$ indeed, the formula looks very confusing to me and I don't know how to read it. Are you sure $a$ is quantified universally ($\forall$) then? Commented Oct 18, 2020 at 19:34
• I don't remember where exactly I got the definition from, but here is a more or less similar one: math.uh.edu/~pwalker/Chap4F09.pdf. Please look at the point 4.2, page 119 at what says "DEFINITION". Though they use "x" as variable there, not "t". A place where time is used is here, for example: sites.oxy.edu/ron/math/341/10/ws/25.pdf. Page 2, on "DEFINITION". Though they don't say my "a" is Real. There is said to be non-negative - not sure why, now that I think about it (possibly because of time, but I'm not getting it right now). Commented Oct 18, 2020 at 19:42
• Just found it. It's here where I took the definition from: personal.psu.edu/sxt104/class/Math251/Notes-LT1.pdf. It's on page 3. Time is used and "a" is any Real number. More confused now, since the variation is more weird than other times. I had not noticed that before you mentioned it. Here they say "a" is any Real number. On the other one I mention on the other comment they say "a" is non-negative and still Time is used. When Real numbers are used, they say "a" is any Real number again. What a mess in my head... Commented Oct 18, 2020 at 19:46
• Sorry, I've been really busy. Just accepted your answer. Though, the definition on the main post is on the site of my last comment before this one. It's translated to "mathmatical form", as on the site it's written with sentences. But I'll go with the ones on the other comment then, as I think you said they're the same (and if they are, I gotta study them a bit so I can see why). Thank you. Commented Oct 23, 2020 at 13:54

There is a standard definition of exponential order used in the theory of Laplace transforms, which is a hypothesis in various theorems about them. This says that a function $$f \colon [ 0 , \infty ) \to \mathbb R$$ is of exponential order if, for some constant $$M \geq 0$$, for some real constant $$K$$, for some real constant $$a$$, for each value $$t \geq M$$, the absolute value of $$f ( t )$$ is bounded above by $$K \mathrm e ^ { a t }$$. With the quantifiers explicit: $$\exists \, M \geq 0 , \; \exists \, K \in \mathbb R , \; \exists \, a \in \mathbb R , \; \forall \, t \geq M , \; | f ( t ) | \leq K \mathrm e ^ { a t } \text .$$ This is the order that the quantifiers must appear in for the theorems in the notes at https://web.archive.org/web/20220408111122/http://www.personal.psu.edu/sxt104/class/Math251/Notes-LT1.pdf to hold. There is some variation possible; for example, since the conclusion can't possibly hold if $$K < 0$$, sometimes you'll see $$K \geq 0$$ required, and even $$K > 0$$ could be required without loss of generality (as could $$a \geq 0$$, $$a > 0$$, or $$M > 0$$). You could even leave $$K$$ out entirely (essentially fixing $$K : = 1$$) by using a larger value of $$a$$. But these definitions are all equivalent.