Complex Analysis: What Does |C| Mean?

[Bachelier]

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

 

[I like Serena]

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 $\mathbb C$.

 

[micromass]

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?

 

[Bachelier]

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.

 

[I like Serena]

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.

 

[Bachelier]

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