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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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    $\begingroup$ A line integral is ultimately a Riemann integral if that is what you're asking. FWIW, I prefer the standard notation over yours. $\endgroup$
    – ShawnD
    Feb 14, 2012 at 17:09
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    $\begingroup$ 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. $\endgroup$
    – J126
    Feb 14, 2012 at 17:14
  • $\begingroup$ I like the notation $\int_{I} f(x)~dx$ fine, but writing out intervals makes my LaTeX look cluttered. $\endgroup$
    – ShawnD
    Feb 14, 2012 at 17:51
  • $\begingroup$ @Shawn: Something I had not thought about. $\endgroup$
    – J126
    Feb 14, 2012 at 18:08
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    $\begingroup$ 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... $\endgroup$
    – Zhen Lin
    Feb 14, 2012 at 19:21

1 Answer 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.

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