# Thread: Infinite width, finite area box function

1. ## Infinite width, finite area box function

Hi! Just out of curiosity, I've been interested in a kind of function $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ with the properties that it's constant valued, but infinitesimally small everywhere in the real domain, and with finite valued integral over its domain. There are many ways to define this function, one being as a box function:

$\displaystyle f(x) \equiv \lim_{w\to\infty}h(x;w) = \left\lbrace\begin{matrix}\frac{1}{2w} &: |x| \leq w\\0 &: |x|>w\end{matrix}\right.$

Is there a name for this function, and what are its potential applications? I suppose it's similar in concept to the Dirac delta function (one definition of the Dirac delta would be simply to take the limit w to 0 in the above definition).

2. ## Re: Infinite width, finite area box function

I've seen

$Rect_w(x) = \begin{cases}\dfrac 1 w &|x| \leq \dfrac w 2\\ 0 &|x| > \dfrac w 2 \end{cases}$

It's used to model a single pulse.

3. ## Re: Infinite width, finite area box function

Yes, I know about the box/rectangular function. The example you gave is simply $\displaystyle Rect_w(x) = h(x;w/2)$, which is a special case of the boxcar function. What I'm interested in is the function in the limit $\displaystyle w\to\infty$, which is no longer useful to model single pulses. Note that you could equally well define the function as a normal distribution with the standard deviation $\displaystyle \sigma\to\infty$, it's still the same function (just as you can define a Dirac delta as a normal distribution with $\displaystyle \sigma\to 0$).

4. ## Re: Infinite width, finite area box function

Originally Posted by sitho
Yes, I know about the box/rectangular function. The example you gave is simply $\displaystyle Rect_w(x) = h(x;w/2)$, which is a special case of the boxcar function. What I'm interested in is the function in the limit $\displaystyle w\to\infty$, which is no longer useful to model single pulses. Note that you could equally well define the function as a normal distribution with the standard deviation $\displaystyle \sigma\to\infty$, it's still the same function (just as you can define a Dirac delta as a normal distribution with $\displaystyle \sigma\to 0$).
and how is this any different from a dirac delta?

5. ## Re: Infinite width, finite area box function

Well, for instance, the integral of the Dirac delta over every integration domain that contains x=0 is 1, whereas the integral of the function that I had in mind is zero for every bounded integration domain.

6. ## Re: Infinite width, finite area box function

Originally Posted by sitho
Hi! Just out of curiosity, I've been interested in a kind of function $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ with the properties that it's constant valued, but infinitesimally small everywhere in the real domain, and with finite valued integral over its domain. There are many ways to define this function, one being as a box function:

$\displaystyle f(x) \equiv \lim_{w\to\infty}h(x;w) = \left\lbrace\begin{matrix}\frac{1}{2w} &: |x| \leq w\\0 &: |x|>w\end{matrix}\right.$

Is there a name for this function, and what are its potential applications? I suppose it's similar in concept to the Dirac delta function (one definition of the Dirac delta would be simply to take the limit w to 0 in the above definition).
Yes, there is a name for that function- it is the 0 function, f(x)= 0 for all x.

7. ## Re: Infinite width, finite area box function

Originally Posted by HallsofIvy
Yes, there is a name for that function- it is the 0 function, f(x)= 0 for all x.
Yes, I can see that for most practical purposes, it simply behaves exactly as f(x) = 0. The exception is upon integration over the whole real domain, where it has a defined finite value, which f(x) = 0 doesn't have in general (unless you clearly define all your zeros and infinities). Maybe that's the answer to my question, that it's a special case of f(x) = 0?

8. ## Re: Infinite width, finite area box function

Then you are NOT talking about the limit in your original post! And f(x)= 0 certainly does have "a defined finite value"- it's 0!

9. ## Re: Infinite width, finite area box function

Originally Posted by HallsofIvy
Then you are NOT talking about the limit in your original post! And f(x)= 0 certainly does have "a defined finite value"- it's 0!
There might be a misunderstanding here. So what I meant in my original post was simply this:

$\displaystyle \int_{-\infty}^\infty dx\:f(x) = \lim_{w\to\infty}\int_{-\infty}^\infty dx\:h(x; w) = \lim_{w\to\infty}\int_{-w}^w dx\:\frac{1}{2w} = \lim_{w\to\infty} 1 = 1$

which is clearly not a standard property of f(x) = 0. Or equivalently expressed as the limit of a Gaussian:

$\displaystyle f(x) = \lim_{\sigma\to\infty}\frac{1}{\sqrt{2\pi}\sigma}e ^{-\frac{x^2}{2\sigma^2}}$
$\displaystyle \int_{-\infty}^\infty dx\:f(x) = \lim_{\sigma\to\infty}\int_{-\infty}^\infty dx\:\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}} = \lim_{\sigma\to\infty} 1 = 1$

P.S. In my previous reply I meant to say "a defined non-zero finite value". I understand that the word "finite" can cause confusion, as it in some contexts (at least for me with a physics background) refers both to non-zero and non-infinite.

10. ## Re: Infinite width, finite area box function

Originally Posted by sitho
Hi! Just out of curiosity, I've been interested in a kind of function $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ with the properties that it's constant valued, but infinitesimally small everywhere in the real domain, and with finite valued integral over its domain. There are many ways to define this function, one being as a box function:

$\displaystyle f(x) \equiv \lim_{w\to\infty}h(x;w) = \left\lbrace\begin{matrix}\frac{1}{2w} &: |x| \leq w\\0 &: |x|>w\end{matrix}\right.$

Is there a name for this function, and what are its potential applications? I suppose it's similar in concept to the Dirac delta function (one definition of the Dirac delta would be simply to take the limit w to 0 in the above definition).
Re-reading this I see that the Dirac delta is sort of the opposite of what you are looking for.

What you're really after is a uniform distribution of infinite width.

This doesn't exist.

Once you let $w$ span the entire real line the integral doesn't converge if $\dfrac{1}{2w}>0$

If $\dfrac{1}{2w}=0$ then you are at what HallsOfIvy mentioned, i.e. $f(x)=0$

11. ## Re: Infinite width, finite area box function

Originally Posted by romsek
Re-reading this I see that the Dirac delta is sort of the opposite of what you are looking for.

What you're really after is a uniform distribution of infinite width.

This doesn't exist.

Once you let $w$ span the entire real line the integral doesn't converge if $\dfrac{1}{2w}>0$

If $\dfrac{1}{2w}=0$ then you are at what HallsOfIvy mentioned, i.e. $f(x)=0$
I would argue for that if one defines both the width and the height of the box function in such a way that its integral is always 1, the integral would in fact converge to 1 also for $\displaystyle w\to\infty$. Or do you see anything wrong with the two derivations in my last post (first with $\displaystyle f(x)$ defined as a box function and then with $\displaystyle f(x)$ as a Gaussian)? I would also say that this infinite width, uniform distribution function exists just as much as the Dirac delta does. Neither this function nor the Dirac delta is ever anything you'd want to work with numerically, but it can still be well defined in an analytical sense, and you can derive any properties from it based on its definition.

12. ## Re: Infinite width, finite area box function

Originally Posted by sitho
I would argue for that if one defines both the width and the height of the box function in such a way that its integral is always 1, the integral would in fact converge to 1 also for $\displaystyle w\to\infty$. Or do you see anything wrong with the two derivations in my last post (first with $\displaystyle f(x)$ defined as a box function and then with $\displaystyle f(x)$ as a Gaussian)? I would also say that this infinite width, uniform distribution function exists just as much as the Dirac delta does. Neither this function nor the Dirac delta is ever anything you'd want to work with numerically, but it can still be well defined in an analytical sense, and you can derive any properties from it based on its definition.
At this point you're just arguing to argue.