The Contour |C|

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  • #1
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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
 

Answers and Replies

  • #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##.
 
  • #3
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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

Do you have a reference for this?
 
  • #4
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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##.

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.
 
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  • #5
I like Serena
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Perhaps it's intended to denote the closure.

##(Interior(C)## not to be confused with ##\mathring{C})##.

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.
 
  • #6
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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.

you got it. Thanks
 

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