Can $Omega$ be replaced with any open set of $Bbb C$ in Theorem 5.2 and Theorem 5.3?












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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?










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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


















0














enter image description here



enter image description here



enter image description here



enter image description here



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?










share|cite|improve this question
























  • where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
    – mathworker21
    yesterday
















0












0








0







enter image description here



enter image description here



enter image description here



enter image description here



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?










share|cite|improve this question















enter image description here



enter image description here



enter image description here



enter image description here



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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  • 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






where does it say $Omega$ is a region? and what is a region? is it just a simply connected, open set?
– mathworker21
yesterday












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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 $Omega$.






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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 $Omega$.






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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 $Omega$.






      share|cite|improve this answer
























        2












        2








        2






        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 $Omega$.






        share|cite|improve this answer












        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 $Omega$.







        share|cite|improve this answer












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        share|cite|improve this answer










        answered yesterday









        Kavi Rama Murthy

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