Difference between ramification point and branch point
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I'm trying to distinguish between the two. According to wiki:
Let $Ω$ be a connected open set in the complex plane $mathbb C$ and $ƒ:Ω → mathbb C$ a holomorphic function. If $ƒ$ is not constant, then the set of the critical points of $ƒ$, that is, the zeros of the derivative $ƒ'(z)$, has no limit point in $Ω$. So each critical point $z_0$ of $ƒ$ lies at the center of a disc $B(z_0,r)$ containing no other critical point of $ƒ$ in its closure.
Let $γ $ be the boundary of $B(z_0,r)$, taken with its positive orientation. The winding number of $ƒ(γ)$ with respect to the point $ƒ(z_0)$ is a positive integer called the ramification index of $z_0$. If the ramification index is greater than 1, then $z_0$ is called a ramification point of ƒ, and the corresponding critical value $ƒ(z_0)$ is called an (algebraic) branch point. Equivalently, $z_0$ is a ramification point if there exists a holomorphic function $φ$ defined in a neighborhood of $z_0$ such that $ƒ(z) = φ(z)(z − z_0)^k$ for some positive integer $k > 1$.
What confuses me are the following:
1) the usage in complex analysis where I think we do not distinguish between the ramification and the branch point.
2) In the wiki definition, what happens if $f(z_0)$ is not defined? Do we still say that $f$ has a ramification point at $z_0$ even though the branch point $f(z_0)$ is undefined?
And another question:
3) Is there an easy computational tool to determine the ramification points? I feel like it has to be related to $f$ having a multiple root at a ramification point, but I don't think that's precise enough.
complex-analysis
asked Nov 23 at 16:55
chhro
1,218310
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up vote
0
down vote
favorite
I'm trying to distinguish between the two. According to wiki:
Let $Ω$ be a connected open set in the complex plane $mathbb C$ and $ƒ:Ω → mathbb C$ a holomorphic function. If $ƒ$ is not constant, then the set of the critical points of $ƒ$, that is, the zeros of the derivative $ƒ'(z)$, has no limit point in $Ω$. So each critical point $z_0$ of $ƒ$ lies at the center of a disc $B(z_0,r)$ containing no other critical point of $ƒ$ in its closure.
Let $γ $ be the boundary of $B(z_0,r)$, taken with its positive orientation. The winding number of $ƒ(γ)$ with respect to the point $ƒ(z_0)$ is a positive integer called the ramification index of $z_0$. If the ramification index is greater than 1, then $z_0$ is called a ramification point of ƒ, and the corresponding critical value $ƒ(z_0)$ is called an (algebraic) branch point. Equivalently, $z_0$ is a ramification point if there exists a holomorphic function $φ$ defined in a neighborhood of $z_0$ such that $ƒ(z) = φ(z)(z − z_0)^k$ for some positive integer $k > 1$.
What confuses me are the following:
1) the usage in complex analysis where I think we do not distinguish between the ramification and the branch point.
2) In the wiki definition, what happens if $f(z_0)$ is not defined? Do we still say that $f$ has a ramification point at $z_0$ even though the branch point $f(z_0)$ is undefined?
And another question:
3) Is there an easy computational tool to determine the ramification points? I feel like it has to be related to $f$ having a multiple root at a ramification point, but I don't think that's precise enough.
complex-analysis
asked Nov 23 at 16:55
chhro
1,218310
If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$
– reuns
Nov 23 at 18:34
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to distinguish between the two. According to wiki:
Let $Ω$ be a connected open set in the complex plane $mathbb C$ and $ƒ:Ω → mathbb C$ a holomorphic function. If $ƒ$ is not constant, then the set of the critical points of $ƒ$, that is, the zeros of the derivative $ƒ'(z)$, has no limit point in $Ω$. So each critical point $z_0$ of $ƒ$ lies at the center of a disc $B(z_0,r)$ containing no other critical point of $ƒ$ in its closure.
Let $γ $ be the boundary of $B(z_0,r)$, taken with its positive orientation. The winding number of $ƒ(γ)$ with respect to the point $ƒ(z_0)$ is a positive integer called the ramification index of $z_0$. If the ramification index is greater than 1, then $z_0$ is called a ramification point of ƒ, and the corresponding critical value $ƒ(z_0)$ is called an (algebraic) branch point. Equivalently, $z_0$ is a ramification point if there exists a holomorphic function $φ$ defined in a neighborhood of $z_0$ such that $ƒ(z) = φ(z)(z − z_0)^k$ for some positive integer $k > 1$.
What confuses me are the following:
1) the usage in complex analysis where I think we do not distinguish between the ramification and the branch point.
2) In the wiki definition, what happens if $f(z_0)$ is not defined? Do we still say that $f$ has a ramification point at $z_0$ even though the branch point $f(z_0)$ is undefined?
And another question:
3) Is there an easy computational tool to determine the ramification points? I feel like it has to be related to $f$ having a multiple root at a ramification point, but I don't think that's precise enough.
complex-analysis
asked Nov 23 at 16:55
chhro
1,218310
I'm trying to distinguish between the two. According to wiki:
Let $Ω$ be a connected open set in the complex plane $mathbb C$ and $ƒ:Ω → mathbb C$ a holomorphic function. If $ƒ$ is not constant, then the set of the critical points of $ƒ$, that is, the zeros of the derivative $ƒ'(z)$, has no limit point in $Ω$. So each critical point $z_0$ of $ƒ$ lies at the center of a disc $B(z_0,r)$ containing no other critical point of $ƒ$ in its closure.
Let $γ $ be the boundary of $B(z_0,r)$, taken with its positive orientation. The winding number of $ƒ(γ)$ with respect to the point $ƒ(z_0)$ is a positive integer called the ramification index of $z_0$. If the ramification index is greater than 1, then $z_0$ is called a ramification point of ƒ, and the corresponding critical value $ƒ(z_0)$ is called an (algebraic) branch point. Equivalently, $z_0$ is a ramification point if there exists a holomorphic function $φ$ defined in a neighborhood of $z_0$ such that $ƒ(z) = φ(z)(z − z_0)^k$ for some positive integer $k > 1$.
What confuses me are the following:
1) the usage in complex analysis where I think we do not distinguish between the ramification and the branch point.
2) In the wiki definition, what happens if $f(z_0)$ is not defined? Do we still say that $f$ has a ramification point at $z_0$ even though the branch point $f(z_0)$ is undefined?
And another question:
3) Is there an easy computational tool to determine the ramification points? I feel like it has to be related to $f$ having a multiple root at a ramification point, but I don't think that's precise enough.
complex-analysis
complex-analysis
asked Nov 23 at 16:55
chhro
1,218310
asked Nov 23 at 16:55
chhro
1,218310
asked Nov 23 at 16:55
chhro
1,218310
asked Nov 23 at 16:55
chhro
1,218310
asked Nov 23 at 16:55
chhro
1,218310
1,218310
If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$
– reuns
Nov 23 at 18:34
add a comment |
If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$
– reuns
Nov 23 at 18:34
If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$
– reuns
Nov 23 at 18:34
If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$
– reuns
Nov 23 at 18:34
add a comment |
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If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$
– reuns
Nov 23 at 18:34