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In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$.

But what is being measured with complex line integration, when you go from a point $z_1$ to a point $z_2$ in the complex plane?

With regular line integration I can see $f(x,y)$ maps $(x,y)$ to a point on the $z$ axis directly above above/below $(x,y)$.

But in the complex case, when you map from the domain $Z$ to the image $W$, you are mapping from $\mathbb{R^2}$ to $\mathbb{R^2}$ ...it is not mapping a point to 'directly above/below'...so I don't have any intuition of what is happening with complex line integration.

  • 1
    It is not just that the function is not mapped "above" or "below", that actually doesn't change the intuition much (you can treat the real and imaginary parts separately if that were the only difference). The main difference is that the line elements $ds$ and $dz$ for the two integrations are very different.2012-02-17
  • 5
    Instead of comparing with the scalar line integral, a much closer relative of the complex line integral is [line integrals for vector fields](http://en.wikipedia.org/wiki/Line_integral#Relation_between_the_line_integral_of_a_vector_field_and_the_complex_line_integral).2012-02-17

3 Answers 3