Is a limit point of branch points a branch point?
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I have come into a discussion with my friends over a complex analysis question:
Is $infty$ a branch point of $log(cos z)$?
I can't get a clear answer to this from the definition of branch points. Maybe it's because I've only learnt about primary complex analysis which defines branch point as 'a point which, if you run $z$ through any contour around it, would change the value of $f(z)$'.
Since $frac{pi}{2}+npi ,forall n in mathbb{Z}$ are branch points of $f(z)=log(cos z)$, $infty$ is a limit point of branch points. Follow the above definition, it seems that any contour around $infty$ would change the value of $f(z)$, but it also seems that this should be the credit of the finite branch points. So, based on rigid definitions, is $infty$ a branch point of $f(z)=log(cos z)$? (Equivalently, is $0$ a branch point of $log(cosfrac{1}{z})$?)
complex-analysis multivalued-functions branch-points
asked Jan 8 at 12:34
ThomasThomas
11
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add a comment |
$begingroup$
I have come into a discussion with my friends over a complex analysis question:
Is $infty$ a branch point of $log(cos z)$?
I can't get a clear answer to this from the definition of branch points. Maybe it's because I've only learnt about primary complex analysis which defines branch point as 'a point which, if you run $z$ through any contour around it, would change the value of $f(z)$'.
Since $frac{pi}{2}+npi ,forall n in mathbb{Z}$ are branch points of $f(z)=log(cos z)$, $infty$ is a limit point of branch points. Follow the above definition, it seems that any contour around $infty$ would change the value of $f(z)$, but it also seems that this should be the credit of the finite branch points. So, based on rigid definitions, is $infty$ a branch point of $f(z)=log(cos z)$? (Equivalently, is $0$ a branch point of $log(cosfrac{1}{z})$?)
complex-analysis multivalued-functions branch-points
asked Jan 8 at 12:34
ThomasThomas
11
$endgroup$
add a comment |
$begingroup$
I have come into a discussion with my friends over a complex analysis question:
Is $infty$ a branch point of $log(cos z)$?
I can't get a clear answer to this from the definition of branch points. Maybe it's because I've only learnt about primary complex analysis which defines branch point as 'a point which, if you run $z$ through any contour around it, would change the value of $f(z)$'.
Since $frac{pi}{2}+npi ,forall n in mathbb{Z}$ are branch points of $f(z)=log(cos z)$, $infty$ is a limit point of branch points. Follow the above definition, it seems that any contour around $infty$ would change the value of $f(z)$, but it also seems that this should be the credit of the finite branch points. So, based on rigid definitions, is $infty$ a branch point of $f(z)=log(cos z)$? (Equivalently, is $0$ a branch point of $log(cosfrac{1}{z})$?)
complex-analysis multivalued-functions branch-points
asked Jan 8 at 12:34
ThomasThomas
11
$endgroup$
I have come into a discussion with my friends over a complex analysis question:
Is $infty$ a branch point of $log(cos z)$?
I can't get a clear answer to this from the definition of branch points. Maybe it's because I've only learnt about primary complex analysis which defines branch point as 'a point which, if you run $z$ through any contour around it, would change the value of $f(z)$'.
Since $frac{pi}{2}+npi ,forall n in mathbb{Z}$ are branch points of $f(z)=log(cos z)$, $infty$ is a limit point of branch points. Follow the above definition, it seems that any contour around $infty$ would change the value of $f(z)$, but it also seems that this should be the credit of the finite branch points. So, based on rigid definitions, is $infty$ a branch point of $f(z)=log(cos z)$? (Equivalently, is $0$ a branch point of $log(cosfrac{1}{z})$?)
complex-analysis multivalued-functions branch-points
complex-analysis multivalued-functions branch-points
asked Jan 8 at 12:34
ThomasThomas
11
asked Jan 8 at 12:34
ThomasThomas
11
asked Jan 8 at 12:34
ThomasThomas
11
asked Jan 8 at 12:34
ThomasThomas
11
asked Jan 8 at 12:34
ThomasThomas
11
11
add a comment |
add a comment |
1 Answer
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By definition, a branch point is an isolated singularity of a multivalued analytic function. That is, for $z_0$ to be a branch point of $f$, there must be a deleted neighbourhood $U$ of $z_0$ such that for each $z in U$, each branch of $f$ is analytic in a neighbourhood of $z$. So no, $infty$ is not a branch point of $log(cos(z))$.
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
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$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
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– Thomas
Jan 8 at 13:42
add a comment |
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1 Answer
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1 Answer
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$begingroup$
By definition, a branch point is an isolated singularity of a multivalued analytic function. That is, for $z_0$ to be a branch point of $f$, there must be a deleted neighbourhood $U$ of $z_0$ such that for each $z in U$, each branch of $f$ is analytic in a neighbourhood of $z$. So no, $infty$ is not a branch point of $log(cos(z))$.
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
$endgroup$
$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
$endgroup$
– Thomas
Jan 8 at 13:42
add a comment |
$begingroup$
By definition, a branch point is an isolated singularity of a multivalued analytic function. That is, for $z_0$ to be a branch point of $f$, there must be a deleted neighbourhood $U$ of $z_0$ such that for each $z in U$, each branch of $f$ is analytic in a neighbourhood of $z$. So no, $infty$ is not a branch point of $log(cos(z))$.
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
$endgroup$
$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
$endgroup$
– Thomas
Jan 8 at 13:42
add a comment |
$begingroup$
By definition, a branch point is an isolated singularity of a multivalued analytic function. That is, for $z_0$ to be a branch point of $f$, there must be a deleted neighbourhood $U$ of $z_0$ such that for each $z in U$, each branch of $f$ is analytic in a neighbourhood of $z$. So no, $infty$ is not a branch point of $log(cos(z))$.
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
$endgroup$
By definition, a branch point is an isolated singularity of a multivalued analytic function. That is, for $z_0$ to be a branch point of $f$, there must be a deleted neighbourhood $U$ of $z_0$ such that for each $z in U$, each branch of $f$ is analytic in a neighbourhood of $z$. So no, $infty$ is not a branch point of $log(cos(z))$.
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
answered Jan 8 at 12:52
Robert IsraelRobert Israel
331k23221477
331k23221477
$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
$endgroup$
– Thomas
Jan 8 at 13:42
add a comment |
$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
$endgroup$
– Thomas
Jan 8 at 13:42
$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
$endgroup$
– Thomas
Jan 8 at 13:42
$begingroup$
Do you have any citation for this 'definition'? Our textbook actually states that a branch point is always a non-isolated singularity since the function's derivative isn't well defined on the branch cut and there are infinitely many points on the branch cut in the branch point's neighborhood.
$endgroup$
– Thomas
Jan 8 at 13:42
add a comment |
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