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Recently I've been writing integrals in the following way, for example

$$\int\limits_{[0,1]} {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt} $$

instead of

$$\int\limits_0^1 {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt} $$

or

$$\int\limits_{\mathbb{R}} {\frac{1}{{1 + {t^2}}}dt} $$

instead of

$$\int\limits_{ - \infty }^\infty {\frac{1}{{1 + {t^2}}}dt} $$

I did this because I thought the new notation highlights the fact that we're integrating over a line interval and not only in the extremes of the interval, so as no to "degrade" the definite integral to

$$\int\limits_a^b {f\left( t \right)dt} = F\left( b \right) - F\left( a \right)$$

Although I know virtually nothing about it, I remembered that in complex integration you integrate over a line, a curve in $\mathbb{R}^2$ as opposed to integrating in $\mathbb{R}$ (an interval). It also rang a bell that integrating over $(a,b)$ is the opposite as integrating over $(b,a)$ (i.e. taking the inverse "path" over the line) and I'm guessing this also happens in complex integration, i.e, the path you take changes the value of the integral.

So that's my doubt, is complex integration a generalization of the common integration in the real domain?

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    A line integral is ultimately a Riemann integral if that is what you're asking. FWIW, I prefer the standard notation over yours.2012-02-14
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    If you've taken a course or read a book on multivariable calculus, then line integrals are definitely a generalization. While @Shawn is welcome to his opinion, I rather like your notation +1.2012-02-14
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    I like the notation $\int_{I} f(x)~dx$ fine, but writing out intervals makes my LaTeX look cluttered.2012-02-14
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    @Shawn: Something I had not thought about.2012-02-14
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    @Shawn What do you mean by cluttered?2012-02-14
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    $\int_a^b$ saves at least 3 characters (maybe more in your actual LaTeX code) over $\int_{(a,b)}$ and (imo) looks cleaner (I don't like having too much as a subscript as I think it looks bad). This is all just aesthetic and there are good reasons to use your notation (see answer below)2012-02-14
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    If you write $\int_X$ and $X$ is _just_ a set, then what you have must be some kind of Lebesgue integral, i.e. unoriented. Philosophically, if one wants to be able to distinguish between $\int_a^b$ and $\int_b^a$ in this notation, one must take $X$ be something with a _chosen_ orientation – in particular, it can't be a set. But it could be a oriented manifold, or it could be a suitable homology class...2012-02-14
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    Way out of my league @ZhenLin.2012-02-14
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    I think the notation is good in the sense that it unifies the various representations of a "definite" integral and makes the special/traditional notation $\int_a^b f$ of an integral over the real line unnecessary. I agree that it looks a bit odd at first, but think I could get used to it.2012-02-14
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    @3Sphere Thanks! Noted.2012-02-14

1 Answers 1

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Indeed, if you parametrize an interval and do a line integral over that "curve", you get the usual integral.

Regarding the notation, your choice is unusual for the basic calculus integral, but is definitely better, in the sense that when you move to Lebesgue integration, you can integrate over sets which are not defined by two endpoints, and you write $$ \int_X\, f $$ Since in Lebesgue integration $X$ may be even non-numerical, in that context the classical notation makes no sense.