Complex line integral

• Apr 11th 2012, 02:24 AM
monster
Complex line integral
I'm doing a question where i have to calculate a the line integral; ∮z .dz over a simple closed contour, which as expected ( By Cauchy) came out as 0.
It's the supplementary questions that are confusing, i'm asked to calculate the integral with z replaced with the conjugate of z, and then to explain the relationship between these two integrals and the integrals over the same contour of just the real(z) and imaginary(z).
can someone shed some light on how they relate.
cheers
• Apr 11th 2012, 07:08 AM
jens
Re: Complex line integral
Try this: if you decompose $z$ into its real and imaginary part, then what are the sum
$\oint z\;dz + \oint \bar{z}\;dz$
and the difference
$\oint z\;dz - \oint \bar{z}\;dz$
equal to?
• Apr 11th 2012, 06:42 PM
monster
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 ?
• Apr 13th 2012, 07:52 AM
jens
Re: Complex line integral
Up to a factor, yes. Just write the integrand $z$ as $\mathfrak{R}\{z\} + i \mathfrak{I}\{z\}$ and use the fact that the integral is a linear operator, in the sense that
$\int \alpha f(x) dx = \alpha \int f(x) dx$
and
$\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$