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I'm sorry if this is a silly question. I've done quite a bit of searching and have not found any definition or name for this symbol/usage, despite immense popularity and convenience. The sources I've checked barely allude to the existence of such a thing, or give vague explanations. It has many varieties and from what I understand can be traced back to early usage of infinitesimals.

I'm talking about, the bar, bracket, or other vertical symbol used to denote a difference of a function between two points, as: $$\int_{a}^{b}{f(x)}=\biggl[ F(x) \biggr]_{a}^{b}=F(x) \Bigg|_{a}^{b}=F(x)\bigg]_{a}^{b}$$ I have seen other usages of such notation (e.g. in the context of partial derivatives, for example, where it can mean 'taken at this point') being defined, but not this usage. Some authors use it, some do not. I see it more often than not in elementary mathematics courses or introductory material, but it appears everywhere.

It does not seem to have an appropriate unicode character, in stark contrast to the dozens of types of spaces, dashes, periods, and other explicitly named vertical bars.

I haven't found anything but cursory mention of it on wikipedia, even in articles talking about the very symbols used to represent it, and despite it appearing on many images and formulas there. At some point I thought I misremembered its popularity, or may have confused it with something else.

Word cannot typeset it, and it doesn't have an exclusive macro in standard latex (as far as I know), although it is often used there as (depending on taste) a large bracket, bar, divider, etc.

What is the name of this notation? Does it have a definition? Is it used, in this specific meaning, anywhere else besides integration? Is there a 'correct' way to typeset it? (e.g. lar or medium square brackets, bar, single square bracket, etc)

I would be thankful if anyone has a source on it.

GregRos
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  • I always though it was an extended use of the restriction notation - the same way that you can put two limits on a $\sum$ or just a set below it. – Alfonso Fernandez Jan 14 '13 at 22:19
  • I’ve never seen a name for any of the variants. I simply use square brackets sized to the expression; I don’t like the one-sided variants at all. – Brian M. Scott Jan 14 '13 at 22:19
  • To me it is just a symbol to denote the (difference of) evaluation of the expression on those values. I type it as \right|_a^b. – Sigur Jan 14 '13 at 22:21
  • Yup that's what it is to everyone. I'm just hoping to find a definition. I'm also hoping for a broad one, actually. Can you have this thing with complex numbers? Anywhere else besides integrals? Who invented it? Is it common in professional writing? – GregRos Jan 14 '13 at 22:38
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    Further to @Sigur's comment, if you're using one of the variants without an opening delimiter, you'll need a \left. to the left of the expression being evaluated. (Note the period after the command, which stands for an empty delimiter.) Thus, your example could be typeset as \int_a^bf(x)\mathrm dx=\left[F(x)\right]_a^b=\left.F(x)\right|_a^b=\left.F(x)\right]_a^b, yielding

    $$ \int_a^bf(x)\mathrm dx=\left[F(x)\right]_a^b=\left.F(x)\right|_a^b=\left.F(x)\right]_a^b $$

     – joriki Jan 14 '13 at 22:59
  • @GregRos, I believe that it is related with the sum word. The symbol $\int$ for integral looks like a S letter. So, you start with a Sum and then pass to an integral $\int$ and finally you evaluate with the vertical bar $\big|$. Did you note some deformation on the letter S? – Sigur Jan 14 '13 at 23:05
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    I am skeptical of the ideas that it’s an extension of the restriction notation or an extreme simplification of the letter S. – Brian M. Scott Jan 14 '13 at 23:21
  • It seems reasonable to suppose it arose as a convenient shorthand with vague resemblance to an integral. I have not seen it in use besides there, and since it is classically given as a single bar or bracket (where logically one would require two) makes this more likely.
    For the record, I usually type it as the first of the variants I presented in the question. I don't like how it looks when the bracket is small, and I don't like the ambiguity of the single bracket. I also don't like the $\rm{dx}$ or the $\int$ but that's another story.     – GregRos Jan 15 '13 at 00:10
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    These are all notations for evaluation. It means evaluated from a to b. – daniel.wright Feb 20 '13 at 03:34
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    Related: http://math.stackexchange.com/questions/52651/what-is-the-name-of-the-vertical-bar – HDE 226868 Jul 11 '15 at 22:16

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