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I have come across some complex integrals with limits when doing questions but have not been able to make sense of what they mean due to my lack of understanding.

We usually define integrals along a path and I have been told that taking a suitable parameterisation is a good way to evaluate these integrals. There are other approaches like using Cauchy's Integral formula etc.

However I saw a definite complex integral written and wanted to make sense of such a thing: what would the following signify? $$\int^i_{1+i} z \, dz$$

My best guess is that this is an integral over the path joining the line $1+i$ to $i$. Could this also represent any path joining these two points in this case? If I were to choose a function that isn't entire like $f(z)=z$ then what would the integral mean?

user258521
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    The exact path doesn't matter if there are no singularities of the integrand. – herb steinberg Feb 24 '18 at 23:01
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    I think this is bad notation: in general, it is not well-defined. But as @herbsteinberg mentioned, this always makes sense for holomorphic functions. – George Feb 24 '18 at 23:02
  • I was evaluating the following integral: $\int_{\gamma} \frac{f(z)}{z-z_0} , dz$ over a finite line from some $-a$ to $a$ along the imaginary line with $f$ holomorphic in the half plane defined by Re$(z) \geq 0$ and $z_0$ taken so that Re$(z_0)>0$, so the function I want to integrate would be holomorphic in some subset of the half plane and thus I can say the integral is independent of the path as you claim? – user258521 Feb 24 '18 at 23:10

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