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The intuition behind the line integral of a function $f$ along a path $\gamma(t)$ on $\mathbb{R}^2$ is clear: it's the area of the surface given by $(t, \gamma(t), f(\gamma(t))$.

I was wondering if there's a good way to think about complex line integration as well, and why one might expect such nice results like the Cauchy integral theorem.

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1. No, the line integral is not equal to the area of that surface. 2. The Cauchy integral theorem comes about because of special properties of complex-differentiable functions, not because of how complex line integration works. – Zhen Lin Oct 1 '11 at 21:52

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