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I feel ashamed to admit that I have taken an applied complex analysis course long before, because honestly I don't think I understand the big picture, which explains why I almost completely forget what I was supposed to learn.

  1. I was wondering what types of integrals are discussed in complex analysis. Are these mostly mentioned ones:

    1. Line integral
    2. integral wrt some Borel measure defined on complex plane, if any?
    3. anything else?
  2. How is Line integral related to integral wrt some Borel measure defined on complex plane, or just on $\mathbb{R}^2$ as which the complex plane is viewed?

Thanks and regards!

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In complex analysis, you study integrals of complex valued functions; because the complex plane is, well, a plane, the analogue of an integral of a real valued function over an interval is the integral of a complex-valued function along a line. – Arturo Magidin Apr 12 '11 at 2:21
@Arturo: Thanks! (1) Or is it more of a analogue of a real-valued function over a line in $\mathbb{R}^2$? (2) in complex analysis, is line integral related to some Lebesgue integral wrt some Borel measure defined on the complex plane? – Tim Apr 12 '11 at 2:27
No, it's not the analogue of a real valued function with domain $\mathbb{R}^2$, because with complex functions the output and the input are both elements of the complex plane. The usual complex integral is the analogue of the Riemann integral, not of the Lebesgue integral. – Arturo Magidin Apr 12 '11 at 2:31
Perhaps you could consider editing your existing questions instead of creating new ones every time a tangent crops up in comments/answers. It would help keep down on site clutter. – Alex Becker Apr 12 '11 at 2:32
@Alex: (1) I was originally going to ask questions about integrals on $\mathbb{C}$ and on $\mathbb{R}^n$ in a same post, but I guessed and still think they are so different that mixing them will not make reply easy. (2) I have no other similar questions to flood the site. – Tim Apr 12 '11 at 2:37

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