Uniform convergence of a sequence of holomorphic functions

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Let $U$ be an open subset of $mathbbC$ and $f_n$ be a sequence of holomorphic functions. Suppose that $f_n$ converges uniformly to a function $f$ on compact subsets of $U$ and that $f$ is not identically zero in $U$ and $f(w)=0$ for some $w in U$.



Prove that there exists $N in mathbbN$ and a sequence $z_n$ such that $f_n(z_n) = 0$ for all $n geq N$ and $lim_n rightarrow inftyz_n = w$.



How can I prove the existence of $z_n$ and $f_n(z_n)$ is exactly 0?
Any idea?




complex-analysis uniform-convergence holomorphic-functions




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  • Feels like an $epsilon/3$ argument to me.
    – Adrian Keister
    Aug 7 at 19:17










  • Hint: try with Hurwitz's theorem.
    – Bob
    Aug 7 at 19:27














up vote
0
down vote

favorite












Let $U$ be an open subset of $mathbbC$ and $f_n$ be a sequence of holomorphic functions. Suppose that $f_n$ converges uniformly to a function $f$ on compact subsets of $U$ and that $f$ is not identically zero in $U$ and $f(w)=0$ for some $w in U$.



Prove that there exists $N in mathbbN$ and a sequence $z_n$ such that $f_n(z_n) = 0$ for all $n geq N$ and $lim_n rightarrow inftyz_n = w$.



How can I prove the existence of $z_n$ and $f_n(z_n)$ is exactly 0?
Any idea?




complex-analysis uniform-convergence holomorphic-functions




share|cite|improve this question



















  • Feels like an $epsilon/3$ argument to me.
    – Adrian Keister
    Aug 7 at 19:17










  • Hint: try with Hurwitz's theorem.
    – Bob
    Aug 7 at 19:27












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $U$ be an open subset of $mathbbC$ and $f_n$ be a sequence of holomorphic functions. Suppose that $f_n$ converges uniformly to a function $f$ on compact subsets of $U$ and that $f$ is not identically zero in $U$ and $f(w)=0$ for some $w in U$.



Prove that there exists $N in mathbbN$ and a sequence $z_n$ such that $f_n(z_n) = 0$ for all $n geq N$ and $lim_n rightarrow inftyz_n = w$.



How can I prove the existence of $z_n$ and $f_n(z_n)$ is exactly 0?
Any idea?




complex-analysis uniform-convergence holomorphic-functions




share|cite|improve this question











Let $U$ be an open subset of $mathbbC$ and $f_n$ be a sequence of holomorphic functions. Suppose that $f_n$ converges uniformly to a function $f$ on compact subsets of $U$ and that $f$ is not identically zero in $U$ and $f(w)=0$ for some $w in U$.



Prove that there exists $N in mathbbN$ and a sequence $z_n$ such that $f_n(z_n) = 0$ for all $n geq N$ and $lim_n rightarrow inftyz_n = w$.



How can I prove the existence of $z_n$ and $f_n(z_n)$ is exactly 0?
Any idea?





complex-analysis uniform-convergence holomorphic-functions





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asked Aug 7 at 19:15









Rachel.L

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  • Feels like an $epsilon/3$ argument to me.
    – Adrian Keister
    Aug 7 at 19:17










  • Hint: try with Hurwitz's theorem.
    – Bob
    Aug 7 at 19:27
















  • Feels like an $epsilon/3$ argument to me.
    – Adrian Keister
    Aug 7 at 19:17










  • Hint: try with Hurwitz's theorem.
    – Bob
    Aug 7 at 19:27















Feels like an $epsilon/3$ argument to me.
– Adrian Keister
Aug 7 at 19:17




Feels like an $epsilon/3$ argument to me.
– Adrian Keister
Aug 7 at 19:17












Hint: try with Hurwitz's theorem.
– Bob
Aug 7 at 19:27




Hint: try with Hurwitz's theorem.
– Bob
Aug 7 at 19:27










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Hint: Let $Gamma$ be a circle around $w$ such that $Gamma$ and its interior are in $U$, and $f_n$ is nonzero on $Gamma$. Then $$dfrac12pi i oint_Gamma dfracf_n'(z); dzf_n(z)$$ is the number of zeros of $f_n$ (counted by multiplicity) inside $Gamma$






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    Hint: Let $Gamma$ be a circle around $w$ such that $Gamma$ and its interior are in $U$, and $f_n$ is nonzero on $Gamma$. Then $$dfrac12pi i oint_Gamma dfracf_n'(z); dzf_n(z)$$ is the number of zeros of $f_n$ (counted by multiplicity) inside $Gamma$






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      up vote
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      down vote













      Hint: Let $Gamma$ be a circle around $w$ such that $Gamma$ and its interior are in $U$, and $f_n$ is nonzero on $Gamma$. Then $$dfrac12pi i oint_Gamma dfracf_n'(z); dzf_n(z)$$ is the number of zeros of $f_n$ (counted by multiplicity) inside $Gamma$






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Hint: Let $Gamma$ be a circle around $w$ such that $Gamma$ and its interior are in $U$, and $f_n$ is nonzero on $Gamma$. Then $$dfrac12pi i oint_Gamma dfracf_n'(z); dzf_n(z)$$ is the number of zeros of $f_n$ (counted by multiplicity) inside $Gamma$






        share|cite|improve this answer













        Hint: Let $Gamma$ be a circle around $w$ such that $Gamma$ and its interior are in $U$, and $f_n$ is nonzero on $Gamma$. Then $$dfrac12pi i oint_Gamma dfracf_n'(z); dzf_n(z)$$ is the number of zeros of $f_n$ (counted by multiplicity) inside $Gamma$







        share|cite|improve this answer













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        share|cite|improve this answer











        answered Aug 7 at 19:48









        Robert Israel

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