How are zeros of a global analytic function defined? Are they isolated?
$begingroup$
According to Wikipedia,
An analytic function in an open set U is called a function element.
A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).
In this regard, I have the following questions:
What is meant by saying that a global analytic function vanishes on an open set in the complex plane?
If the above question makes sense, then does it imply that the global analytic function is identically zero?
Some references for the above would be very helpful.
Thanks in advance.
complex-analysis analysis reference-request definition
asked Dec 28 '18 at 6:47
RamaRama
113
$endgroup$
|
show 2 more comments
$begingroup$
According to Wikipedia,
An analytic function in an open set U is called a function element.
A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).
In this regard, I have the following questions:
What is meant by saying that a global analytic function vanishes on an open set in the complex plane?
If the above question makes sense, then does it imply that the global analytic function is identically zero?
Some references for the above would be very helpful.
Thanks in advance.
complex-analysis analysis reference-request definition
asked Dec 28 '18 at 6:47
RamaRama
113
$endgroup$
1
$begingroup$
In what context have you heard/read "a global analytic function vanishes on an open set in the complex plane"? Please edit the question to include the details.
$endgroup$
– Shaun
Dec 28 '18 at 6:51
$begingroup$
I've deleted my answer, @Rama; I'm sorry: it was wrong due to a technicality. Please edit your question to include the definition of a global analytic function to avoid such a confusion in future.
$endgroup$
– Shaun
Dec 28 '18 at 7:10
1
$begingroup$
@Shaun : No problem. Many thanks for your immediate and kind response.
$endgroup$
– Rama
Dec 28 '18 at 7:13
$begingroup$
You're welcome. I'd like to apologise to @EricWofsey, too.
$endgroup$
– Shaun
Dec 28 '18 at 7:14
$begingroup$
Thank you for editing the question, @Rama. However, it'd help to have a link to the specific Wikipedia article you are referring to. Type[text](link)to have a hyperlink with URL "link" and text "text".
$endgroup$
– Shaun
Dec 28 '18 at 7:16
|
show 2 more comments
$begingroup$
According to Wikipedia,
An analytic function in an open set U is called a function element.
A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).
In this regard, I have the following questions:
What is meant by saying that a global analytic function vanishes on an open set in the complex plane?
If the above question makes sense, then does it imply that the global analytic function is identically zero?
Some references for the above would be very helpful.
Thanks in advance.
complex-analysis analysis reference-request definition
asked Dec 28 '18 at 6:47
RamaRama
113
$endgroup$
According to Wikipedia,
An analytic function in an open set U is called a function element.
A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).
In this regard, I have the following questions:
What is meant by saying that a global analytic function vanishes on an open set in the complex plane?
If the above question makes sense, then does it imply that the global analytic function is identically zero?
Some references for the above would be very helpful.
Thanks in advance.
complex-analysis analysis reference-request definition
complex-analysis analysis reference-request definition
asked Dec 28 '18 at 6:47
RamaRama
113
asked Dec 28 '18 at 6:47
RamaRama
113
edited Dec 28 '18 at 7:12
Rama
asked Dec 28 '18 at 6:47
RamaRama
113
asked Dec 28 '18 at 6:47
RamaRama
113
asked Dec 28 '18 at 6:47
RamaRama
113
113
1
$begingroup$
In what context have you heard/read "a global analytic function vanishes on an open set in the complex plane"? Please edit the question to include the details.
$endgroup$
– Shaun
Dec 28 '18 at 6:51
$begingroup$
I've deleted my answer, @Rama; I'm sorry: it was wrong due to a technicality. Please edit your question to include the definition of a global analytic function to avoid such a confusion in future.
$endgroup$
– Shaun
Dec 28 '18 at 7:10
1
$begingroup$
@Shaun : No problem. Many thanks for your immediate and kind response.
$endgroup$
– Rama
Dec 28 '18 at 7:13
$begingroup$
You're welcome. I'd like to apologise to @EricWofsey, too.
$endgroup$
– Shaun
Dec 28 '18 at 7:14
$begingroup$
Thank you for editing the question, @Rama. However, it'd help to have a link to the specific Wikipedia article you are referring to. Type[text](link)to have a hyperlink with URL "link" and text "text".
$endgroup$
– Shaun
Dec 28 '18 at 7:16
|
show 2 more comments
1
$begingroup$
In what context have you heard/read "a global analytic function vanishes on an open set in the complex plane"? Please edit the question to include the details.
$endgroup$
– Shaun
Dec 28 '18 at 6:51
$begingroup$
I've deleted my answer, @Rama; I'm sorry: it was wrong due to a technicality. Please edit your question to include the definition of a global analytic function to avoid such a confusion in future.
$endgroup$
– Shaun
Dec 28 '18 at 7:10
1
$begingroup$
@Shaun : No problem. Many thanks for your immediate and kind response.
$endgroup$
– Rama
Dec 28 '18 at 7:13
$begingroup$
You're welcome. I'd like to apologise to @EricWofsey, too.
$endgroup$
– Shaun
Dec 28 '18 at 7:14
$begingroup$
Thank you for editing the question, @Rama. However, it'd help to have a link to the specific Wikipedia article you are referring to. Type[text](link)to have a hyperlink with URL "link" and text "text".
$endgroup$
– Shaun
Dec 28 '18 at 7:16
1
1
$begingroup$
In what context have you heard/read "a global analytic function vanishes on an open set in the complex plane"? Please edit the question to include the details.
$endgroup$
– Shaun
Dec 28 '18 at 6:51
$begingroup$
In what context have you heard/read "a global analytic function vanishes on an open set in the complex plane"? Please edit the question to include the details.
$endgroup$
– Shaun
Dec 28 '18 at 6:51
$begingroup$
I've deleted my answer, @Rama; I'm sorry: it was wrong due to a technicality. Please edit your question to include the definition of a global analytic function to avoid such a confusion in future.
$endgroup$
– Shaun
Dec 28 '18 at 7:10
$begingroup$
I've deleted my answer, @Rama; I'm sorry: it was wrong due to a technicality. Please edit your question to include the definition of a global analytic function to avoid such a confusion in future.
$endgroup$
– Shaun
Dec 28 '18 at 7:10
1
1
$begingroup$
@Shaun : No problem. Many thanks for your immediate and kind response.
$endgroup$
– Rama
Dec 28 '18 at 7:13
$begingroup$
@Shaun : No problem. Many thanks for your immediate and kind response.
$endgroup$
– Rama
Dec 28 '18 at 7:13
$begingroup$
You're welcome. I'd like to apologise to @EricWofsey, too.
$endgroup$
– Shaun
Dec 28 '18 at 7:14
$begingroup$
You're welcome. I'd like to apologise to @EricWofsey, too.
$endgroup$
– Shaun
Dec 28 '18 at 7:14
$begingroup$
Thank you for editing the question, @Rama. However, it'd help to have a link to the specific Wikipedia article you are referring to. Type
[text](link) to have a hyperlink with URL "link" and text "text".$endgroup$
– Shaun
Dec 28 '18 at 7:16
$begingroup$
Thank you for editing the question, @Rama. However, it'd help to have a link to the specific Wikipedia article you are referring to. Type
[text](link) to have a hyperlink with URL "link" and text "text".$endgroup$
– Shaun
Dec 28 '18 at 7:16
|
show 2 more comments
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$begingroup$
In what context have you heard/read "a global analytic function vanishes on an open set in the complex plane"? Please edit the question to include the details.
$endgroup$
– Shaun
Dec 28 '18 at 6:51
$begingroup$
I've deleted my answer, @Rama; I'm sorry: it was wrong due to a technicality. Please edit your question to include the definition of a global analytic function to avoid such a confusion in future.
$endgroup$
– Shaun
Dec 28 '18 at 7:10
1
$begingroup$
@Shaun : No problem. Many thanks for your immediate and kind response.
$endgroup$
– Rama
Dec 28 '18 at 7:13
$begingroup$
You're welcome. I'd like to apologise to @EricWofsey, too.
$endgroup$
– Shaun
Dec 28 '18 at 7:14
$begingroup$
Thank you for editing the question, @Rama. However, it'd help to have a link to the specific Wikipedia article you are referring to. Type
[text](link)to have a hyperlink with URL "link" and text "text".$endgroup$
– Shaun
Dec 28 '18 at 7:16