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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor series in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable (complex-)differentiable at every point of $S^1$. (where the differentiability is global at a point instead of only along certain directions in $\overline\Delta$.)
Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ uniform or absolute on $\overline\Delta$? (Note."uniform" does not imply "absolute".)
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor series in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable at every point of $S^1$. (where the differentiability is global at some a point instead of only along certain directions in $\overline\Delta$.)
Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ uniform or absolute on $\overline\Delta$? (Note."uniform" does not imply "absolute".)
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor seris series in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable at every point of $S^1$. (where the differentiability is global at some point instead of only along certain directions in $\overline\Delta$.)
Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ is uniform or absolute on $\overline\Delta$? (Note."uniform" doesnot does not imply "absolute".)
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive the closed disk $\epsilon$. \overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor seris in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable at every point of $S^1$. (where the differentiability is global at some point instead of only along certain directions in $\overline\Delta$.)
Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ is uniform or absolute on the closed disk $\overline\Delta$? (Note."uniform" doesnot imply "absolute".)
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ \overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Let $f(z)=\sum a_nz^n$ be its Taylor seris in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable at every point of $S^1$.
Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$? (Note."uniform" doesnot imply "absolute".)
Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ is uniform or absolute on the closed disk $\overline\Delta$? (Note."uniform" doesnot imply "absolute".)
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Let $f(z)=\sum a_nz^n$ be its Taylor seris in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable at every point of $S^1$.
Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$? (Note."uniform" doesnot imply "absolute".)
Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ is uniform or absolute on the closed disk $\overline\Delta$?
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Let $f(z)=\sum a_nz^n$ be its Taylor seris in $\Delta$ which has $R=1$ as its radius of convergence.
We further assume that $f(z)$ is differentiable at every point of $S^1$.
I have the following questionSuch a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice:
1. Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
2. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Let $f(z)=\sum a_nz^n$ be its Taylor seris in $\Delta$ which has $R=1$ as its radius of convergence.
I have the following question.We further assume that $f(z)$ is differentiable at every point of $S^1$.
I have the following question.
Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice:
1. Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
2. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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Is there an On analytic function differentiable on the circle of convergence of its Taylor series?
For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Let $f(z)=\sum a_nz^n$ be its Taylor seris in $\Delta$ which has $R=1$ as its radius of convergence.
I have the following question.
Question 1. Is there such a function
We further assume that $f(z)$ which is differentiable at every point of $S^1$?
If the answer to S^1$.
Question 1is yes, let $f$ be such a function.
Question 2. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice 1. Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.
- There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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Is there an analytic function differentiable on the circle of convergence of its natural boundaryTaylor series?
For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Suppose that the unit circle Let $S^1$ is the nuatural (f(z)=\sum a_nz^n$ be its Taylor seris in analytic sense) boundary of $f(z)$. \Delta$ which has $R=1$ as its radius of convergence.
I have the following question.
Question 1. Is there such a function $f(z)$ which is differentiable at every point of $S^1$?
If the answer to Question 1 is yes, let $f$ be such a functionand $f=\sum a_nz^n$ its Taylor seris in $\Delta$. .
Question 2. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$. We further assumet that $f$ is <1$ and continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Suppose that the unit circle $S^1$ is the nuatural (in analytic sense) boundary of $f(z)$. I have the following question.
Question 1. Is there such a $f(z)$ which is differentiable at every point of $S^1$?
If the answer to Question 1 is yes, let $f$ be such a function and $f=\sum a_nz^n$ its Taylor seris in $\Delta$.
Question 2. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$. We further assumet that $f$ is continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Suppose that the unit circle $S^1$ is the nuatural (in analytic sense) boundary of $f(z)$. I have the following question.
Question 1. Is there such a $f(z)$ which is differentiable at every point of $S^1$?
If the answer to Question 1 is yes. Let , let $f$ be such a function and $f=\sum a_nz^n$ its Taylor seris in $\Delta$.
Question 2. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice. There is a function $f$ analytic in $\Delta$ and continous in $\Delta(\epsilon)$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this):
http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series
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Is there an analytic function differentiable on its natural boundary?
For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$. We further assumet that $f$ is continous in $\Delta(\epsilon): |z|<1+\epsilon$ for some positive $\epsilon$. Suppose that the unit circle $S^1$ is the nuatural (in analytic sense) boundary of $f(z)$. I have the following question.
Question 1. Is there such a $f(z)$ which is differentiable at every point of $S^1$?
If the answer to Question 1 is yes. Let $f$ be such a function and $f=\sum a_nz^n$ its Taylor seris in $\Delta$.
Question 2. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?
Notice.
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