Re: Complex line integral

Try this: if you decompose $\displaystyle z$ into its real and imaginary part, then what are the sum

$\displaystyle \oint z\;dz + \oint \bar{z}\;dz$

and the difference

$\displaystyle \oint z\;dz - \oint \bar{z}\;dz$

equal to?

Re: Complex line integral

I think i see, is i that the countour integral of the real part plus the contour integral of the imag part = contour integral of z, and the difference of them = contour integral of conjugate of z ?

Re: Complex line integral

Up to a factor, yes. Just write the integrand $\displaystyle z$ as $\displaystyle \mathfrak{R}\{z\} + i \mathfrak{I}\{z\}$ and use the fact that the integral is a linear operator, in the sense that

$\displaystyle \int \alpha f(x) dx = \alpha \int f(x) dx$

and

$\displaystyle \int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$