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 $z_0$.










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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














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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 $z_0$.










share|cite|improve this question























  • 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












up vote
0
down vote

favorite









up vote
0
down vote

favorite











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 $z_0$.










share|cite|improve this question















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






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edited Sep 9 at 9:45

























asked Sep 9 at 9:32









user270331

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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
















  • 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










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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.






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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.






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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.






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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.






        share|cite|improve this answer












        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.







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        answered Sep 9 at 9:56









        Frieder Jäckel

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