singularities( Essential and removable) of a complex function. The 2019 Stack Overflow Developer Survey Results Are InComplex analysis removable singularitiesRemovable singularities of a holomorphic functionRemovable Singularities for a exponential type functionProof Essential Singularities are IsolatedHow does squaring a function affect it's removable singularities?Essential singularities and polesCauchy Goursat and removable singularitiesRemovable singularities and an entire functionapplication of Riemann's theorem on removable singularitiesCan a meromorphic function have removable singularities?
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singularities( Essential and removable) of a complex function.
The 2019 Stack Overflow Developer Survey Results Are InComplex analysis removable singularitiesRemovable singularities of a holomorphic functionRemovable Singularities for a exponential type functionProof Essential Singularities are IsolatedHow does squaring a function affect it's removable singularities?Essential singularities and polesCauchy Goursat and removable singularitiesRemovable singularities and an entire functionapplication of Riemann's theorem on removable singularitiesCan a meromorphic function have removable singularities?
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I have some doubts on this question.
I think the first and second are false. As for the others I'm not sure.
complex-analysis
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
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add a comment |
$begingroup$
I have some doubts on this question.
I think the first and second are false. As for the others I'm not sure.
complex-analysis
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
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The first one is clearly true.
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– Saucy O'Path
Apr 7 at 21:57
$begingroup$
you are right. i meant the second and fourth.
$endgroup$
– Luís Cruz
Apr 7 at 21:59
add a comment |
$begingroup$
I have some doubts on this question.
I think the first and second are false. As for the others I'm not sure.
complex-analysis
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
$endgroup$
I have some doubts on this question.
I think the first and second are false. As for the others I'm not sure.
complex-analysis
complex-analysis
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
asked Apr 7 at 21:52
Luís CruzLuís Cruz
183
183
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The first one is clearly true.
$endgroup$
– Saucy O'Path
Apr 7 at 21:57
$begingroup$
you are right. i meant the second and fourth.
$endgroup$
– Luís Cruz
Apr 7 at 21:59
add a comment |
$begingroup$
The first one is clearly true.
$endgroup$
– Saucy O'Path
Apr 7 at 21:57
$begingroup$
you are right. i meant the second and fourth.
$endgroup$
– Luís Cruz
Apr 7 at 21:59
$begingroup$
The first one is clearly true.
$endgroup$
– Saucy O'Path
Apr 7 at 21:57
$begingroup$
The first one is clearly true.
$endgroup$
– Saucy O'Path
Apr 7 at 21:57
$begingroup$
you are right. i meant the second and fourth.
$endgroup$
– Luís Cruz
Apr 7 at 21:59
$begingroup$
you are right. i meant the second and fourth.
$endgroup$
– Luís Cruz
Apr 7 at 21:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Actually, the first one is true: if the Laurent series of $f$ at $z_0$ has infinitely many non-zero terms of the type $a_n(z-z_0)^n$ with $n<0$, then the same thing occurs with $(z-z_0)^2f(z)$.
But you are right about the secnd one: it is false.
The third one is true: you are multiplying $f(z)$ by an analytic function whose domain contains $z_0$: Therefore, the essential singularity at $z_0$ remains as such.
And the fourth one is false, of course.
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
add a comment |
Your Answer
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
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active
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votes
$begingroup$
Actually, the first one is true: if the Laurent series of $f$ at $z_0$ has infinitely many non-zero terms of the type $a_n(z-z_0)^n$ with $n<0$, then the same thing occurs with $(z-z_0)^2f(z)$.
But you are right about the secnd one: it is false.
The third one is true: you are multiplying $f(z)$ by an analytic function whose domain contains $z_0$: Therefore, the essential singularity at $z_0$ remains as such.
And the fourth one is false, of course.
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
add a comment |
$begingroup$
Actually, the first one is true: if the Laurent series of $f$ at $z_0$ has infinitely many non-zero terms of the type $a_n(z-z_0)^n$ with $n<0$, then the same thing occurs with $(z-z_0)^2f(z)$.
But you are right about the secnd one: it is false.
The third one is true: you are multiplying $f(z)$ by an analytic function whose domain contains $z_0$: Therefore, the essential singularity at $z_0$ remains as such.
And the fourth one is false, of course.
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
add a comment |
$begingroup$
Actually, the first one is true: if the Laurent series of $f$ at $z_0$ has infinitely many non-zero terms of the type $a_n(z-z_0)^n$ with $n<0$, then the same thing occurs with $(z-z_0)^2f(z)$.
But you are right about the secnd one: it is false.
The third one is true: you are multiplying $f(z)$ by an analytic function whose domain contains $z_0$: Therefore, the essential singularity at $z_0$ remains as such.
And the fourth one is false, of course.
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
Actually, the first one is true: if the Laurent series of $f$ at $z_0$ has infinitely many non-zero terms of the type $a_n(z-z_0)^n$ with $n<0$, then the same thing occurs with $(z-z_0)^2f(z)$.
But you are right about the secnd one: it is false.
The third one is true: you are multiplying $f(z)$ by an analytic function whose domain contains $z_0$: Therefore, the essential singularity at $z_0$ remains as such.
And the fourth one is false, of course.
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
answered Apr 7 at 21:58
José Carlos SantosJosé Carlos Santos
174k23133242
174k23133242
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
add a comment |
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
$begingroup$
Thank you for your help. I meant that the second and fourth are false (my bad).
$endgroup$
– Luís Cruz
Apr 7 at 22:04
add a comment |
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$begingroup$
The first one is clearly true.
$endgroup$
– Saucy O'Path
Apr 7 at 21:57
$begingroup$
you are right. i meant the second and fourth.
$endgroup$
– Luís Cruz
Apr 7 at 21:59