# What is the difference between following approaches to line integrals?

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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• 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.
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
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