Application of winding number and the roots of complex polynomial from a non simple closed cuvreroots of...
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Application of winding number and the roots of complex polynomial from a non simple closed cuvre
roots of $f(z)=z^4+8z^3+3z^2+8z+3=0$ in the right half planeLemma in deriving the winding number?Homotopy invariance of winding number in complex analysis.Defining the winding number for a general curveWinding number of a simple closed curve in a plane.Is there a closed loop in the complex plane such that for any given integer $x$, I can find a point inside the loop that has winding number $x$?closed path, winding number, Jordan contourWinding number in physics and mathematicsconnection between winding number and topological degreeWhen is the winding number undefined
$begingroup$
There is a formula for the simple closed curve $gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the closed curve $gamma(t)$. This is just a direct result from the argument principle.
Now I want to ask what if the closed curve $gamma(t)$ is NOT simple anymore which means it may have self-intersections. What will the winding number $w(p(gamma),(0,0))$ be? I think it should be still related to the roots of $p(z)$
complex-analysis polynomials curves winding-number
edited yesterday
Martin R
29.7k33558
asked yesterday
Phoebe Phoebe
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Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
$begingroup$
There is a formula for the simple closed curve $gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the closed curve $gamma(t)$. This is just a direct result from the argument principle.
Now I want to ask what if the closed curve $gamma(t)$ is NOT simple anymore which means it may have self-intersections. What will the winding number $w(p(gamma),(0,0))$ be? I think it should be still related to the roots of $p(z)$
complex-analysis polynomials curves winding-number
edited yesterday
Martin R
29.7k33558
asked yesterday
Phoebe Phoebe
61
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
There is a formula for the simple closed curve $gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the closed curve $gamma(t)$. This is just a direct result from the argument principle.
Now I want to ask what if the closed curve $gamma(t)$ is NOT simple anymore which means it may have self-intersections. What will the winding number $w(p(gamma),(0,0))$ be? I think it should be still related to the roots of $p(z)$
complex-analysis polynomials curves winding-number
edited yesterday
Martin R
29.7k33558
asked yesterday
Phoebe Phoebe
61
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
There is a formula for the simple closed curve $gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the closed curve $gamma(t)$. This is just a direct result from the argument principle.
Now I want to ask what if the closed curve $gamma(t)$ is NOT simple anymore which means it may have self-intersections. What will the winding number $w(p(gamma),(0,0))$ be? I think it should be still related to the roots of $p(z)$
complex-analysis polynomials curves winding-number
complex-analysis polynomials curves winding-number
edited yesterday
Martin R
29.7k33558
asked yesterday
Phoebe Phoebe
61
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Martin R
29.7k33558
asked yesterday
Phoebe Phoebe
61
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Martin R
29.7k33558
edited yesterday
Martin R
29.7k33558
edited yesterday
Martin R
29.7k33558
29.7k33558
asked yesterday
Phoebe Phoebe
61
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Phoebe Phoebe
61
asked yesterday
Phoebe Phoebe
61
61
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Phoebe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
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oldest
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$begingroup$
The winding number of $p circ gamma $ with respect to $0$ is
$$
I(p circ gamma,0) = frac{1}{2pi i}int_{p circ gamma} frac{dw}{w}
= frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
$$
and that can be computed with the Residue theorem. If $a_1, ldots, a_m$ are the distinct roots of $p$ with multiplicities $k_1, ldots ,k_m$, then
$$
frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
= sum_{j=1}^m I(gamma, a_j) operatorname{Res}(frac{p'}{p}, a_j)
, .
$$
It is relatively easy to compute that the residue of $p'/p$ at a $k$-fold zero of $p$ is $k$, so that
$$
I(p circ gamma,0)= sum_{j=1}^m I(gamma, a_j) k_j , .
$$
This is the general formula.
For a simple positively oriented closed curve $gamma$ the winding numbers $I(gamma, a_j)$ are either zero or one, and the sum reduces to
$$
I(p circ gamma,0) = sum_{a_j text{ inside } gamma} k_j
$$
answered yesterday
Martin RMartin R
29.7k33558
$endgroup$
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The winding number of $p circ gamma $ with respect to $0$ is
$$
I(p circ gamma,0) = frac{1}{2pi i}int_{p circ gamma} frac{dw}{w}
= frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
$$
and that can be computed with the Residue theorem. If $a_1, ldots, a_m$ are the distinct roots of $p$ with multiplicities $k_1, ldots ,k_m$, then
$$
frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
= sum_{j=1}^m I(gamma, a_j) operatorname{Res}(frac{p'}{p}, a_j)
, .
$$
It is relatively easy to compute that the residue of $p'/p$ at a $k$-fold zero of $p$ is $k$, so that
$$
I(p circ gamma,0)= sum_{j=1}^m I(gamma, a_j) k_j , .
$$
This is the general formula.
For a simple positively oriented closed curve $gamma$ the winding numbers $I(gamma, a_j)$ are either zero or one, and the sum reduces to
$$
I(p circ gamma,0) = sum_{a_j text{ inside } gamma} k_j
$$
answered yesterday
Martin RMartin R
29.7k33558
$endgroup$
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
add a comment |
$begingroup$
The winding number of $p circ gamma $ with respect to $0$ is
$$
I(p circ gamma,0) = frac{1}{2pi i}int_{p circ gamma} frac{dw}{w}
= frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
$$
and that can be computed with the Residue theorem. If $a_1, ldots, a_m$ are the distinct roots of $p$ with multiplicities $k_1, ldots ,k_m$, then
$$
frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
= sum_{j=1}^m I(gamma, a_j) operatorname{Res}(frac{p'}{p}, a_j)
, .
$$
It is relatively easy to compute that the residue of $p'/p$ at a $k$-fold zero of $p$ is $k$, so that
$$
I(p circ gamma,0)= sum_{j=1}^m I(gamma, a_j) k_j , .
$$
This is the general formula.
For a simple positively oriented closed curve $gamma$ the winding numbers $I(gamma, a_j)$ are either zero or one, and the sum reduces to
$$
I(p circ gamma,0) = sum_{a_j text{ inside } gamma} k_j
$$
answered yesterday
Martin RMartin R
29.7k33558
$endgroup$
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
add a comment |
$begingroup$
The winding number of $p circ gamma $ with respect to $0$ is
$$
I(p circ gamma,0) = frac{1}{2pi i}int_{p circ gamma} frac{dw}{w}
= frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
$$
and that can be computed with the Residue theorem. If $a_1, ldots, a_m$ are the distinct roots of $p$ with multiplicities $k_1, ldots ,k_m$, then
$$
frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
= sum_{j=1}^m I(gamma, a_j) operatorname{Res}(frac{p'}{p}, a_j)
, .
$$
It is relatively easy to compute that the residue of $p'/p$ at a $k$-fold zero of $p$ is $k$, so that
$$
I(p circ gamma,0)= sum_{j=1}^m I(gamma, a_j) k_j , .
$$
This is the general formula.
For a simple positively oriented closed curve $gamma$ the winding numbers $I(gamma, a_j)$ are either zero or one, and the sum reduces to
$$
I(p circ gamma,0) = sum_{a_j text{ inside } gamma} k_j
$$
answered yesterday
Martin RMartin R
29.7k33558
$endgroup$
The winding number of $p circ gamma $ with respect to $0$ is
$$
I(p circ gamma,0) = frac{1}{2pi i}int_{p circ gamma} frac{dw}{w}
= frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
$$
and that can be computed with the Residue theorem. If $a_1, ldots, a_m$ are the distinct roots of $p$ with multiplicities $k_1, ldots ,k_m$, then
$$
frac{1}{2pi i}int_gamma frac{p'(z)}{p(z)} , dz
= sum_{j=1}^m I(gamma, a_j) operatorname{Res}(frac{p'}{p}, a_j)
, .
$$
It is relatively easy to compute that the residue of $p'/p$ at a $k$-fold zero of $p$ is $k$, so that
$$
I(p circ gamma,0)= sum_{j=1}^m I(gamma, a_j) k_j , .
$$
This is the general formula.
For a simple positively oriented closed curve $gamma$ the winding numbers $I(gamma, a_j)$ are either zero or one, and the sum reduces to
$$
I(p circ gamma,0) = sum_{a_j text{ inside } gamma} k_j
$$
answered yesterday
Martin RMartin R
29.7k33558
edited yesterday
answered yesterday
Martin RMartin R
29.7k33558
answered yesterday
Martin RMartin R
29.7k33558
answered yesterday
Martin RMartin R
29.7k33558
29.7k33558
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
add a comment |
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
The formula for a simple closed curve is missing a sign depending on the orientation of $gamma$.
$endgroup$
– Paul Frost
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
$begingroup$
@PaulFrost: You are right, thanks.
$endgroup$
– Martin R
yesterday
add a comment |
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