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Why is $z_0=0$ not an essential singularity of $tan(1/z)$ What is this singularity?
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Since for any $epsilon>0~exists |z|=|frac{2}{(2k+1)pi}|<epsilon$ for some $k$ large enough such that $tan(1/z)=pminfty$ there cannot be a Laurent series at $0$.
Does there need to be a Laurent series converging in a neighborhood of $0$ for it to be an essential singularity? Clearly for any $R>0$ the series does not converge in ${|z|<R}$ in this case.
What do we call this type of singularity?
complex-analysis laurent-series singularity
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
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add a comment |
$begingroup$
Since for any $epsilon>0~exists |z|=|frac{2}{(2k+1)pi}|<epsilon$ for some $k$ large enough such that $tan(1/z)=pminfty$ there cannot be a Laurent series at $0$.
Does there need to be a Laurent series converging in a neighborhood of $0$ for it to be an essential singularity? Clearly for any $R>0$ the series does not converge in ${|z|<R}$ in this case.
What do we call this type of singularity?
complex-analysis laurent-series singularity
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
$endgroup$
3
$begingroup$
May be related: en.wikipedia.org/wiki/…
$endgroup$
– Nick
Jan 2 at 7:45
add a comment |
$begingroup$
Since for any $epsilon>0~exists |z|=|frac{2}{(2k+1)pi}|<epsilon$ for some $k$ large enough such that $tan(1/z)=pminfty$ there cannot be a Laurent series at $0$.
Does there need to be a Laurent series converging in a neighborhood of $0$ for it to be an essential singularity? Clearly for any $R>0$ the series does not converge in ${|z|<R}$ in this case.
What do we call this type of singularity?
complex-analysis laurent-series singularity
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
$endgroup$
Since for any $epsilon>0~exists |z|=|frac{2}{(2k+1)pi}|<epsilon$ for some $k$ large enough such that $tan(1/z)=pminfty$ there cannot be a Laurent series at $0$.
Does there need to be a Laurent series converging in a neighborhood of $0$ for it to be an essential singularity? Clearly for any $R>0$ the series does not converge in ${|z|<R}$ in this case.
What do we call this type of singularity?
complex-analysis laurent-series singularity
complex-analysis laurent-series singularity
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
edited Jan 2 at 7:43
John Cataldo
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
asked Jan 2 at 7:30
John CataldoJohn Cataldo
1,1071216
1,1071216
3
$begingroup$
May be related: en.wikipedia.org/wiki/…
$endgroup$
– Nick
Jan 2 at 7:45
add a comment |
3
$begingroup$
May be related: en.wikipedia.org/wiki/…
$endgroup$
– Nick
Jan 2 at 7:45
3
3
$begingroup$
May be related: en.wikipedia.org/wiki/…
$endgroup$
– Nick
Jan 2 at 7:45
$begingroup$
May be related: en.wikipedia.org/wiki/…
$endgroup$
– Nick
Jan 2 at 7:45
add a comment |
1 Answer
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$0$ is not an isolated singularity. There is no disk around $0$ In which the function is analytic except for the origin.
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
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add a comment |
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$begingroup$
$0$ is not an isolated singularity. There is no disk around $0$ In which the function is analytic except for the origin.
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
$endgroup$
add a comment |
$begingroup$
$0$ is not an isolated singularity. There is no disk around $0$ In which the function is analytic except for the origin.
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
$endgroup$
add a comment |
$begingroup$
$0$ is not an isolated singularity. There is no disk around $0$ In which the function is analytic except for the origin.
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
$endgroup$
$0$ is not an isolated singularity. There is no disk around $0$ In which the function is analytic except for the origin.
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
answered Jan 2 at 8:13
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
52.9k32055
add a comment |
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May be related: en.wikipedia.org/wiki/…
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– Nick
Jan 2 at 7:45