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What is the difference between following approaches concerning line integrals:

  1. first approach is for complex function with parametrization $\gamma(t) = \cos t+i\sin t$ (Line_integral :: Example from Wikipedia).

  2. Second approach is for $f(x,y)$ with parametrization $x=\cos t, y=\sin t$ (Line_integral :: Example from khanacademy.org).

Yes complex function $f(z)=1/z$ is $f: C \rightarrow C$ while the other is $f:R^2 \rightarrow R$, but what other differences there are or are they same thing said in different method?

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Let me add one more kind of line integration to make the difference clearer:

  • Line integral of a scalar function. This is your second approach. There is not much going on here. An intuitive description of the integral would be to sum the function values encountered along the curve.
  • Line integral of a vector function. Take a vector function with components $(f(x,y),g(x,y))$. Then the integral is defined by $ \int \left( f(x(t),y(t))x'(t)+g(x(t),y(t))y'(t) \right) {\mathrm{d}}t. $ A rough description would be to sum the tangential component of the vector field $(f,g)$ along the curve.
  • Line integral of a complex function. Take a complex function $f(x,y)+ig(x,y)$. Then the integral is $ \int \left( f(x(t),y(t))x'(t)-g(x(t),y(t))y'(t) \right) {\mathrm{d}}t + i\int \left( f(x(t),y(t))y'(t)+g(x(t),y(t))x'(t) \right) {\mathrm{d}}t. $ As you see, this is somewhat similar to the vector integral, in the sense that the integral depends on the interaction between the components of the function and the curve in a nontrivial way. However, this is "more scalar" than the vector integral because of the complex multiplication, in the same way complex numbers are "more scalar" than just two dimensional vectors.
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    @alvoutila: By the way your complex line integral is wrong. There should be no absolute value sign. These are definitions, so they are not really derived. For motivations you can read any introductory complex analysis book. One good book is Gamelin's *Complex analysis*.2012-08-07