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my question is quite simple and might be a duplicate though I couldn't find one. Is there an accepted meaning for the notation $$\int_{z_1}^{z_2}$$ With $z_1,z_2\in\mathbb{C}$? Is the standard to just integrate over the line connecting the two points?

Thanks.

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    $\begingroup$ This notation is absolutely terrible. Sure, it could mean the straight line segment, but sometimes this may not make sense, for instance, $\int_{-1}^1\frac{1}{z}\,dz$ makes no sense if we look at the straight line segment, but it does make sense if you prescribe for example a semicircular arc which avoids $0$. On occasion I've seen this notation used for when the integrand is an exact differential form (hence path-independent). But even then I always like to indicate the path used. $\endgroup$ – peek-a-boo Jan 16 at 23:04
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Yes. When your function $f$ is "path-independent" in the domain, this notation is sometimes used for $$ \int_\gamma f(z)dz=\int_{z_1}^{z_2} f(z)dz $$ where $z_1$ is the initial point of $\gamma$ and $z_2$ the final point.

However, when $f$ is not path-independent, one usually use the left-hand side and give the explicit description of $\gamma$.

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