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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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Usually you don't use $h(z)= f(x,y)+ig(x,y)$ but for example $f(z)=\frac{1}{z}$. Why you use $f(x,y)+ig(x,y)$ notation? Could you tell site where that Line integral of a complex function is with a proof. I have a site where this integral is $\int_{\gamma} f(z)dz = \int_a^b f(\gamma(t))\gamma'(t)dt$(en.wikipedia.org/wiki/Line_integral#Complex_line_integral) Can you explain how this differs from your equation( line integral of complex function) –  alvoutila Aug 5 '12 at 16:09
@alvoutila: there is no difference. Of course you don't use that notation. I used it to compare with the vector integral. However, note that here we have a general complex-valued function rather than a holomorphic function. General complex-valued functions are not much different than vector-valued functions in 2D. –  timur Aug 5 '12 at 23:01
Can you give me references. Where you can find construction/deduction for this complex line integral and by the way why one puts $\int_{\gamma}f(x,y)dS = \int_{a}^{b}f(x(t),y(t))dS$, where $dS=\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}$ in scalar line integral, while in complex line integral it is $\int_{\gamma}f(z)dz = \int_a^b f(\gamma(t))|\gamma'(t)|dt$? So how these are derived? Can you find any site where these have been derived. –  alvoutila Aug 7 '12 at 15:35
@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. –  timur Aug 7 '12 at 15:41

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