Proof that for every Meromorphic function, it`s derevative is also Meromorphic
$begingroup$
for $f(z)$ Meromorphic we know that $f(z)$ on $Omega$ - an open set ,is Holomorphic on $Omega/A$ when $A$ is discrete set of the poles of $f(z)$.
I know that $f'(z)$ is also Holomorphic on $Omega/A$ because $f(z)$ analytic on this set.
Would it be enough to write $f(z)=frac {g(z)}{h(z)}$ and $g(z),h(z)$ are holomorphic functions on $Bbb C$, and then $f'(z)=frac {g'(z)h(z)-g(z)h'(z)}{(h(z))^2}$
so we know that the set of zeros of $f(z), f'(z)$ denominator is the same, so we know this set is A and it's elements are poles for $f(z)$ and $f'(z)$?
complex-analysis meromorphic-functions
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
$endgroup$
add a comment |
$begingroup$
for $f(z)$ Meromorphic we know that $f(z)$ on $Omega$ - an open set ,is Holomorphic on $Omega/A$ when $A$ is discrete set of the poles of $f(z)$.
I know that $f'(z)$ is also Holomorphic on $Omega/A$ because $f(z)$ analytic on this set.
Would it be enough to write $f(z)=frac {g(z)}{h(z)}$ and $g(z),h(z)$ are holomorphic functions on $Bbb C$, and then $f'(z)=frac {g'(z)h(z)-g(z)h'(z)}{(h(z))^2}$
so we know that the set of zeros of $f(z), f'(z)$ denominator is the same, so we know this set is A and it's elements are poles for $f(z)$ and $f'(z)$?
complex-analysis meromorphic-functions
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
$endgroup$
add a comment |
$begingroup$
for $f(z)$ Meromorphic we know that $f(z)$ on $Omega$ - an open set ,is Holomorphic on $Omega/A$ when $A$ is discrete set of the poles of $f(z)$.
I know that $f'(z)$ is also Holomorphic on $Omega/A$ because $f(z)$ analytic on this set.
Would it be enough to write $f(z)=frac {g(z)}{h(z)}$ and $g(z),h(z)$ are holomorphic functions on $Bbb C$, and then $f'(z)=frac {g'(z)h(z)-g(z)h'(z)}{(h(z))^2}$
so we know that the set of zeros of $f(z), f'(z)$ denominator is the same, so we know this set is A and it's elements are poles for $f(z)$ and $f'(z)$?
complex-analysis meromorphic-functions
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
$endgroup$
for $f(z)$ Meromorphic we know that $f(z)$ on $Omega$ - an open set ,is Holomorphic on $Omega/A$ when $A$ is discrete set of the poles of $f(z)$.
I know that $f'(z)$ is also Holomorphic on $Omega/A$ because $f(z)$ analytic on this set.
Would it be enough to write $f(z)=frac {g(z)}{h(z)}$ and $g(z),h(z)$ are holomorphic functions on $Bbb C$, and then $f'(z)=frac {g'(z)h(z)-g(z)h'(z)}{(h(z))^2}$
so we know that the set of zeros of $f(z), f'(z)$ denominator is the same, so we know this set is A and it's elements are poles for $f(z)$ and $f'(z)$?
complex-analysis meromorphic-functions
complex-analysis meromorphic-functions
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
asked Dec 4 '18 at 11:50
Daniel VainshteinDaniel Vainshtein
19011
19011
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes, that would be enough. When you differentiate a meromorphic function, your are not adding any new poles, and the original poels remain poles (that is, they don't become essential singularities).
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
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1 Answer
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1 Answer
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$begingroup$
Yes, that would be enough. When you differentiate a meromorphic function, your are not adding any new poles, and the original poels remain poles (that is, they don't become essential singularities).
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
add a comment |
$begingroup$
Yes, that would be enough. When you differentiate a meromorphic function, your are not adding any new poles, and the original poels remain poles (that is, they don't become essential singularities).
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
add a comment |
$begingroup$
Yes, that would be enough. When you differentiate a meromorphic function, your are not adding any new poles, and the original poels remain poles (that is, they don't become essential singularities).
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
Yes, that would be enough. When you differentiate a meromorphic function, your are not adding any new poles, and the original poels remain poles (that is, they don't become essential singularities).
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
answered Dec 4 '18 at 11:52
José Carlos SantosJosé Carlos Santos
158k22126228
158k22126228
add a comment |
add a comment |
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