Can $Omega$ be replaced with any open set of $Bbb C$ in Theorem 5.2 and Theorem 5.3?
The above pictures are from Stein's "Complex Analysis".
In Theorem 5.2 and Theorem 5.3, $Omega$ is a region, I think it can be replaced with any open set of $Bbb C$, can't it?
complex-analysis
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Born to be proud
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The above pictures are from Stein's "Complex Analysis".
In Theorem 5.2 and Theorem 5.3, $Omega$ is a region, I think it can be replaced with any open set of $Bbb C$, can't it?
complex-analysis
asked yesterday
Born to be proud
782510
where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday
add a comment |
The above pictures are from Stein's "Complex Analysis".
In Theorem 5.2 and Theorem 5.3, $Omega$ is a region, I think it can be replaced with any open set of $Bbb C$, can't it?
complex-analysis
asked yesterday
Born to be proud
782510
The above pictures are from Stein's "Complex Analysis".
In Theorem 5.2 and Theorem 5.3, $Omega$ is a region, I think it can be replaced with any open set of $Bbb C$, can't it?
complex-analysis
complex-analysis
asked yesterday
Born to be proud
782510
asked yesterday
Born to be proud
782510
edited yesterday
asked yesterday
Born to be proud
782510
asked yesterday
Born to be proud
782510
asked yesterday
Born to be proud
782510
782510
where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday
add a comment |
where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday
where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday
where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday
add a comment |
1 Answer
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Even if the theorems are stated for connected open sets you can apply them to connected components and conclude that they are valid for any open set $Omega$. The conclusions are 'local' results so connectedness is irrelevant. In fact the proofs themselves work for any open set $Omega$. Note that any compact subset of $Omega$ can be covered by a finite number of disks whose closures are contained in $Omega$.
answered yesterday
Kavi Rama Murthy
49.2k31854
add a comment |
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1 Answer
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Even if the theorems are stated for connected open sets you can apply them to connected components and conclude that they are valid for any open set $Omega$. The conclusions are 'local' results so connectedness is irrelevant. In fact the proofs themselves work for any open set $Omega$. Note that any compact subset of $Omega$ can be covered by a finite number of disks whose closures are contained in $Omega$.
answered yesterday
Kavi Rama Murthy
49.2k31854
add a comment |
Even if the theorems are stated for connected open sets you can apply them to connected components and conclude that they are valid for any open set $Omega$. The conclusions are 'local' results so connectedness is irrelevant. In fact the proofs themselves work for any open set $Omega$. Note that any compact subset of $Omega$ can be covered by a finite number of disks whose closures are contained in $Omega$.
answered yesterday
Kavi Rama Murthy
49.2k31854
add a comment |
Even if the theorems are stated for connected open sets you can apply them to connected components and conclude that they are valid for any open set $Omega$. The conclusions are 'local' results so connectedness is irrelevant. In fact the proofs themselves work for any open set $Omega$. Note that any compact subset of $Omega$ can be covered by a finite number of disks whose closures are contained in $Omega$.
answered yesterday
Kavi Rama Murthy
49.2k31854
Even if the theorems are stated for connected open sets you can apply them to connected components and conclude that they are valid for any open set $Omega$. The conclusions are 'local' results so connectedness is irrelevant. In fact the proofs themselves work for any open set $Omega$. Note that any compact subset of $Omega$ can be covered by a finite number of disks whose closures are contained in $Omega$.
answered yesterday
Kavi Rama Murthy
49.2k31854
answered yesterday
Kavi Rama Murthy
49.2k31854
answered yesterday
Kavi Rama Murthy
49.2k31854
answered yesterday
Kavi Rama Murthy
49.2k31854
49.2k31854
add a comment |
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where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday