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The Contour |C|

  1. Mar 16, 2013 #1
    I have a question pertaining to Complex Analysis.

    We define a contour ##C## as a piecewise smooth arc.

    For a variable ##z \in \mathbb{C}## I have seen the notation of a contour ##|C|##. It is sometimes defined as ##|C| := z([a,b])## where ##[a,b]## is a closed interval.

    Should I read this as the parametrization of the contour ##C## between ##a, \ b##?

    Or does ##|C|## have a different meaning ##w.r.t.## contours?

    Thanks
     
  2. jcsd
  3. Mar 17, 2013 #2

    I like Serena

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    The notation ##|C|## indicates a norm that has a curve as its argument.
    The obvious norm is the length of the curve.

    Your definition looks faulty.
    If your curve ##C## is a continuously differentiable function ##z: [a,b] \to \mathbb C##, then:

    ##C=z([a,b])##

    ##|C|=\int_a^b |z'(t)|dt##

    Note that the norm of ##C## is deduced from the norm on ##\mathbb C##.
     
  4. Mar 17, 2013 #3

    micromass

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    Do you have a reference for this?
     
  5. Mar 17, 2013 #4
    Thank you I.L.S. :)

    I clearly see your point, but I think the speaker in this case gave a different definition to ##|C|## to that of the length, as he has defined the length by ##L(C) = \int_a^b \ |z'(t)| \mathrm{d}t##.

    I think he meant that ##|C|## is the curve by itself without the interior as he sometimes used the notation: ##|C| \ \bigcup \ Interior(C)##.

    ##(Interior(C)## not to be confused with ##C^\circ)##.

    But the problem is that he used ##C \ \bigcup \ Interior(C)## as well. So I think it was just a forgetful omission in the latter.

    Since I searched for the symbol ##|C|## and it is not existent in any textbooks, it must thus be a nomenclature he decided to create.
     
    Last edited: Mar 17, 2013
  6. Mar 17, 2013 #5

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    Perhaps it's intended to denote the closure.

    What is the difference between these two?

    I haven't seen ##\mathring{C}## before, although I know that ##C^\circ## is one of the notations for the interior.
     
  7. Mar 17, 2013 #6
    you got it. Thanks
     
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