If $pcirc f$ has a pole at $z_0$ then so does $f$, where $p$ is a polynomial
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Let $f : U$ $z_0 to C$ be a holomorphic function, where $U$ is an open disk and let $p$ be a non-constant polynomial. Show that the singularity of $f$ at $z_0$
is a pole of $f$ iff it is a pole of $p circ f$.
I have proved the 'only if' part of the above question but i am having trouble in proving the 'if' part. Suppose $pcirc f$ has a pole at $z_0$ i.e. There is a holomorphic function $g: Uto C$ with $g(z_0) neq 0$ such that $$pcirc f = fracg(z)(z-z_0)^n$$
So, how should I conclude that f has a pole at $z_0$.
complex-analysis complex-geometry
asked Sep 9 at 9:32
user270331
896
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Let $f : U$ $z_0 to C$ be a holomorphic function, where $U$ is an open disk and let $p$ be a non-constant polynomial. Show that the singularity of $f$ at $z_0$
is a pole of $f$ iff it is a pole of $p circ f$.
I have proved the 'only if' part of the above question but i am having trouble in proving the 'if' part. Suppose $pcirc f$ has a pole at $z_0$ i.e. There is a holomorphic function $g: Uto C$ with $g(z_0) neq 0$ such that $$pcirc f = fracg(z)(z-z_0)^n$$
So, how should I conclude that f has a pole at $z_0$.
complex-analysis complex-geometry
asked Sep 9 at 9:32
user270331
896
is $U$ an open disk? The result seems untrue otherwise
– user254433
Sep 9 at 9:38
1
sorry, yes U is a open disk.
– user270331
Sep 9 at 9:41
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up vote
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Let $f : U$ $z_0 to C$ be a holomorphic function, where $U$ is an open disk and let $p$ be a non-constant polynomial. Show that the singularity of $f$ at $z_0$
is a pole of $f$ iff it is a pole of $p circ f$.
I have proved the 'only if' part of the above question but i am having trouble in proving the 'if' part. Suppose $pcirc f$ has a pole at $z_0$ i.e. There is a holomorphic function $g: Uto C$ with $g(z_0) neq 0$ such that $$pcirc f = fracg(z)(z-z_0)^n$$
So, how should I conclude that f has a pole at $z_0$.
complex-analysis complex-geometry
asked Sep 9 at 9:32
user270331
896
Let $f : U$ $z_0 to C$ be a holomorphic function, where $U$ is an open disk and let $p$ be a non-constant polynomial. Show that the singularity of $f$ at $z_0$
is a pole of $f$ iff it is a pole of $p circ f$.
I have proved the 'only if' part of the above question but i am having trouble in proving the 'if' part. Suppose $pcirc f$ has a pole at $z_0$ i.e. There is a holomorphic function $g: Uto C$ with $g(z_0) neq 0$ such that $$pcirc f = fracg(z)(z-z_0)^n$$
So, how should I conclude that f has a pole at $z_0$.
complex-analysis complex-geometry
complex-analysis complex-geometry
asked Sep 9 at 9:32
user270331
896
asked Sep 9 at 9:32
user270331
896
edited Sep 9 at 9:45
asked Sep 9 at 9:32
user270331
896
asked Sep 9 at 9:32
user270331
896
asked Sep 9 at 9:32
user270331
896
896
is $U$ an open disk? The result seems untrue otherwise
– user254433
Sep 9 at 9:38
1
sorry, yes U is a open disk.
– user270331
Sep 9 at 9:41
add a comment |Â
is $U$ an open disk? The result seems untrue otherwise
– user254433
Sep 9 at 9:38
1
sorry, yes U is a open disk.
– user270331
Sep 9 at 9:41
is $U$ an open disk? The result seems untrue otherwise
– user254433
Sep 9 at 9:38
is $U$ an open disk? The result seems untrue otherwise
– user254433
Sep 9 at 9:38
1
1
sorry, yes U is a open disk.
– user270331
Sep 9 at 9:41
sorry, yes U is a open disk.
– user270331
Sep 9 at 9:41
add a comment |Â
1 Answer
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Hint: $f$ has a pole at $z_0$ if and only if $lim_zto z_0|f(z)|=infty$ and $f$ has an essential season singularity at $z_0$ if and only if the image of $B(z_0,delta)setminus z_0$ under $f$ is dense in $mathbbC$ for all $delta$ small enough.
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint: $f$ has a pole at $z_0$ if and only if $lim_zto z_0|f(z)|=infty$ and $f$ has an essential season singularity at $z_0$ if and only if the image of $B(z_0,delta)setminus z_0$ under $f$ is dense in $mathbbC$ for all $delta$ small enough.
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
add a comment |Â
up vote
0
down vote
Hint: $f$ has a pole at $z_0$ if and only if $lim_zto z_0|f(z)|=infty$ and $f$ has an essential season singularity at $z_0$ if and only if the image of $B(z_0,delta)setminus z_0$ under $f$ is dense in $mathbbC$ for all $delta$ small enough.
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Hint: $f$ has a pole at $z_0$ if and only if $lim_zto z_0|f(z)|=infty$ and $f$ has an essential season singularity at $z_0$ if and only if the image of $B(z_0,delta)setminus z_0$ under $f$ is dense in $mathbbC$ for all $delta$ small enough.
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
Hint: $f$ has a pole at $z_0$ if and only if $lim_zto z_0|f(z)|=infty$ and $f$ has an essential season singularity at $z_0$ if and only if the image of $B(z_0,delta)setminus z_0$ under $f$ is dense in $mathbbC$ for all $delta$ small enough.
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
answered Sep 9 at 9:56
Frieder Jäckel
1,059213
1,059213
add a comment |Â
add a comment |Â
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is $U$ an open disk? The result seems untrue otherwise
– user254433
Sep 9 at 9:38
1
sorry, yes U is a open disk.
– user270331
Sep 9 at 9:41