Distributional Laplacian of $log|F(z)|$ Where F is Entire












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Let $f(z) = log|F(z)|$, where $F: mathbb{C} rightarrow mathbb{C}$ is entire. Then $f$ defines a distribution on $mathbb{R}^2$, and we want to show that its distributional Laplacian is



$$Delta f = sum_{n in I} 2 pi , d_n , delta_{z_n}$$



where ${z_n mid n in I}$ are the zeros of $F$, which have respective degree (multiplicity) $d_n$.





This is what I have got so far (though I suspect that there could be a completely different approach which is easier, so solutions that don't use this work are more than welcome!):



Let $U_varepsilon = mathbb{R}^2 backslash left( bigcuplimits_{n in I} B(z_n , varepsilon) right)$.



Then, for a test function $varphi in D(mathbb{R}^2)$,



$$leftlangle f,Deltavarphirightrangle=lim_{varepsilonrightarrow0^+}int_{U_varepsilon} fDeltavarphi dx,$$
and by Green's second identity
$$int_{U_varepsilon} fDeltavarphi dx=int_{U_varepsilon} varphiDelta f dx+int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds-int_{partial U_varepsilon}varphifrac{partial f}{partial n} ds.$$



The first integral on the RHS vanishes since $f = Re(log(F))$, so is harmonic on $U_varepsilon$ where $log(F)$ is holomorphic.



We can obviously split $partial U_varepsilon$ into circles around each singularity of $f$.



This is where I get stuck. How can we deal with the terms $ffrac{partialvarphi}{partial n}$ and $frac{partial f}{partial n}$.





Any suggestions would be appreciated!










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  • $begingroup$
    Can't we test $varphi$ that vanish on the boundary of $U_epsilon$?
    $endgroup$
    – mathworker21
    Dec 30 '18 at 13:38










  • $begingroup$
    @mathworker21 I'm not too sure what you mean... To show the equality, we need to consider all test functions...
    $endgroup$
    – John Don
    Dec 30 '18 at 16:40
















0












$begingroup$


Let $f(z) = log|F(z)|$, where $F: mathbb{C} rightarrow mathbb{C}$ is entire. Then $f$ defines a distribution on $mathbb{R}^2$, and we want to show that its distributional Laplacian is



$$Delta f = sum_{n in I} 2 pi , d_n , delta_{z_n}$$



where ${z_n mid n in I}$ are the zeros of $F$, which have respective degree (multiplicity) $d_n$.





This is what I have got so far (though I suspect that there could be a completely different approach which is easier, so solutions that don't use this work are more than welcome!):



Let $U_varepsilon = mathbb{R}^2 backslash left( bigcuplimits_{n in I} B(z_n , varepsilon) right)$.



Then, for a test function $varphi in D(mathbb{R}^2)$,



$$leftlangle f,Deltavarphirightrangle=lim_{varepsilonrightarrow0^+}int_{U_varepsilon} fDeltavarphi dx,$$
and by Green's second identity
$$int_{U_varepsilon} fDeltavarphi dx=int_{U_varepsilon} varphiDelta f dx+int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds-int_{partial U_varepsilon}varphifrac{partial f}{partial n} ds.$$



The first integral on the RHS vanishes since $f = Re(log(F))$, so is harmonic on $U_varepsilon$ where $log(F)$ is holomorphic.



We can obviously split $partial U_varepsilon$ into circles around each singularity of $f$.



This is where I get stuck. How can we deal with the terms $ffrac{partialvarphi}{partial n}$ and $frac{partial f}{partial n}$.





Any suggestions would be appreciated!










share|cite|improve this question











$endgroup$












  • $begingroup$
    Can't we test $varphi$ that vanish on the boundary of $U_epsilon$?
    $endgroup$
    – mathworker21
    Dec 30 '18 at 13:38










  • $begingroup$
    @mathworker21 I'm not too sure what you mean... To show the equality, we need to consider all test functions...
    $endgroup$
    – John Don
    Dec 30 '18 at 16:40














0












0








0


1



$begingroup$


Let $f(z) = log|F(z)|$, where $F: mathbb{C} rightarrow mathbb{C}$ is entire. Then $f$ defines a distribution on $mathbb{R}^2$, and we want to show that its distributional Laplacian is



$$Delta f = sum_{n in I} 2 pi , d_n , delta_{z_n}$$



where ${z_n mid n in I}$ are the zeros of $F$, which have respective degree (multiplicity) $d_n$.





This is what I have got so far (though I suspect that there could be a completely different approach which is easier, so solutions that don't use this work are more than welcome!):



Let $U_varepsilon = mathbb{R}^2 backslash left( bigcuplimits_{n in I} B(z_n , varepsilon) right)$.



Then, for a test function $varphi in D(mathbb{R}^2)$,



$$leftlangle f,Deltavarphirightrangle=lim_{varepsilonrightarrow0^+}int_{U_varepsilon} fDeltavarphi dx,$$
and by Green's second identity
$$int_{U_varepsilon} fDeltavarphi dx=int_{U_varepsilon} varphiDelta f dx+int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds-int_{partial U_varepsilon}varphifrac{partial f}{partial n} ds.$$



The first integral on the RHS vanishes since $f = Re(log(F))$, so is harmonic on $U_varepsilon$ where $log(F)$ is holomorphic.



We can obviously split $partial U_varepsilon$ into circles around each singularity of $f$.



This is where I get stuck. How can we deal with the terms $ffrac{partialvarphi}{partial n}$ and $frac{partial f}{partial n}$.





Any suggestions would be appreciated!










share|cite|improve this question











$endgroup$




Let $f(z) = log|F(z)|$, where $F: mathbb{C} rightarrow mathbb{C}$ is entire. Then $f$ defines a distribution on $mathbb{R}^2$, and we want to show that its distributional Laplacian is



$$Delta f = sum_{n in I} 2 pi , d_n , delta_{z_n}$$



where ${z_n mid n in I}$ are the zeros of $F$, which have respective degree (multiplicity) $d_n$.





This is what I have got so far (though I suspect that there could be a completely different approach which is easier, so solutions that don't use this work are more than welcome!):



Let $U_varepsilon = mathbb{R}^2 backslash left( bigcuplimits_{n in I} B(z_n , varepsilon) right)$.



Then, for a test function $varphi in D(mathbb{R}^2)$,



$$leftlangle f,Deltavarphirightrangle=lim_{varepsilonrightarrow0^+}int_{U_varepsilon} fDeltavarphi dx,$$
and by Green's second identity
$$int_{U_varepsilon} fDeltavarphi dx=int_{U_varepsilon} varphiDelta f dx+int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds-int_{partial U_varepsilon}varphifrac{partial f}{partial n} ds.$$



The first integral on the RHS vanishes since $f = Re(log(F))$, so is harmonic on $U_varepsilon$ where $log(F)$ is holomorphic.



We can obviously split $partial U_varepsilon$ into circles around each singularity of $f$.



This is where I get stuck. How can we deal with the terms $ffrac{partialvarphi}{partial n}$ and $frac{partial f}{partial n}$.





Any suggestions would be appreciated!







integration complex-analysis distribution-theory harmonic-functions laplacian






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edited Dec 31 '18 at 16:24







John Don

















asked Dec 30 '18 at 13:06









John DonJohn Don

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366115












  • $begingroup$
    Can't we test $varphi$ that vanish on the boundary of $U_epsilon$?
    $endgroup$
    – mathworker21
    Dec 30 '18 at 13:38










  • $begingroup$
    @mathworker21 I'm not too sure what you mean... To show the equality, we need to consider all test functions...
    $endgroup$
    – John Don
    Dec 30 '18 at 16:40


















  • $begingroup$
    Can't we test $varphi$ that vanish on the boundary of $U_epsilon$?
    $endgroup$
    – mathworker21
    Dec 30 '18 at 13:38










  • $begingroup$
    @mathworker21 I'm not too sure what you mean... To show the equality, we need to consider all test functions...
    $endgroup$
    – John Don
    Dec 30 '18 at 16:40
















$begingroup$
Can't we test $varphi$ that vanish on the boundary of $U_epsilon$?
$endgroup$
– mathworker21
Dec 30 '18 at 13:38




$begingroup$
Can't we test $varphi$ that vanish on the boundary of $U_epsilon$?
$endgroup$
– mathworker21
Dec 30 '18 at 13:38












$begingroup$
@mathworker21 I'm not too sure what you mean... To show the equality, we need to consider all test functions...
$endgroup$
– John Don
Dec 30 '18 at 16:40




$begingroup$
@mathworker21 I'm not too sure what you mean... To show the equality, we need to consider all test functions...
$endgroup$
– John Don
Dec 30 '18 at 16:40










1 Answer
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About the term $int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds$, note that $partial U_epsilon = bigcup_n partial B(z_n,epsilon)$. Fix $n$ and observe that
$$
F(z) = (z-z_n)^d G(z)
$$
where $G(z)$ does not vanish near $z=z_n$. This implies
$$
f(z) = dlog |z-z_n| + log|G(z)|=dlog epsilon + log|G(z)|
$$
on $|z-z_n|=epsilon$. Note that $log |G(z)|$ is a continuous harmonic function and hence $|log |G(z)||leq C$ for some $C>0$. Now, we see that
$$
int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =dlogepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle ds +int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle ds.
$$
From
$$
dBig|logepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle dsBig| = d logepsilon^{-1} int_{|z-z_n|=epsilon} |nabla varphi| dsleq d logepsilon^{-1} cdot 2piepsilon cdotsup|nabla varphi|to 0,
$$
and
$$
Big|int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle dsBig|leq 2pi Cepsilon sup|nablavarphi|to 0,
$$
we get $$
lim_{epsilonto 0}int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =0.
$$
This holds for all $n$, and we conclude $lim_{epsilonto 0}int_{partial U_epsilon} ffrac{partialvarphi}{partial n}ds=0.$



Next, note that $$frac{partial f(z_n + re^{itheta})}{partial n} = frac{partial}{partial r} f(z_n + re^{itheta})= frac{partial}{partial r}[dlog r + log |G(z_n+re^{itheta})|] = frac{d}{r}+partial_rlog |G(z_n+re^{itheta})|. $$ This is because the unit outward normal vector $n$ of the circle $|z-z_n|=r$ at $z = z_n +re^{itheta}$ is just $e^{itheta}$. Now observe that
$$
int_{|z-z_n|=epsilon} varphifrac{partial f}{partial n}ds = frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds + int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)ds .
$$
The first term is
$$
frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds= frac{d}{epsilon}int_0^{2pi} varphi(z_n + epsilon e^{itheta} )epsilon dtheta =dint_0^{2pi} varphi(z_n + epsilon e^{itheta} )dtheta to 2pi d varphi(z_n).
$$
The integrand in the second term is a continuous function and thus
$$
Big|int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)dsBig| leq sup |varphi|cdot sup|partial_r log |G||cdot 2piepsilon to 0.
$$
Summing up over all $n$, we see that
$$
lim_{epsilonto 0}int_{U_varepsilon} fDeltavarphi dx = sum_n 2pi d_n varphi(z_n),
$$
as desired.






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    $begingroup$

    About the term $int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds$, note that $partial U_epsilon = bigcup_n partial B(z_n,epsilon)$. Fix $n$ and observe that
    $$
    F(z) = (z-z_n)^d G(z)
    $$
    where $G(z)$ does not vanish near $z=z_n$. This implies
    $$
    f(z) = dlog |z-z_n| + log|G(z)|=dlog epsilon + log|G(z)|
    $$
    on $|z-z_n|=epsilon$. Note that $log |G(z)|$ is a continuous harmonic function and hence $|log |G(z)||leq C$ for some $C>0$. Now, we see that
    $$
    int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =dlogepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle ds +int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle ds.
    $$
    From
    $$
    dBig|logepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle dsBig| = d logepsilon^{-1} int_{|z-z_n|=epsilon} |nabla varphi| dsleq d logepsilon^{-1} cdot 2piepsilon cdotsup|nabla varphi|to 0,
    $$
    and
    $$
    Big|int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle dsBig|leq 2pi Cepsilon sup|nablavarphi|to 0,
    $$
    we get $$
    lim_{epsilonto 0}int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =0.
    $$
    This holds for all $n$, and we conclude $lim_{epsilonto 0}int_{partial U_epsilon} ffrac{partialvarphi}{partial n}ds=0.$



    Next, note that $$frac{partial f(z_n + re^{itheta})}{partial n} = frac{partial}{partial r} f(z_n + re^{itheta})= frac{partial}{partial r}[dlog r + log |G(z_n+re^{itheta})|] = frac{d}{r}+partial_rlog |G(z_n+re^{itheta})|. $$ This is because the unit outward normal vector $n$ of the circle $|z-z_n|=r$ at $z = z_n +re^{itheta}$ is just $e^{itheta}$. Now observe that
    $$
    int_{|z-z_n|=epsilon} varphifrac{partial f}{partial n}ds = frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds + int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)ds .
    $$
    The first term is
    $$
    frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds= frac{d}{epsilon}int_0^{2pi} varphi(z_n + epsilon e^{itheta} )epsilon dtheta =dint_0^{2pi} varphi(z_n + epsilon e^{itheta} )dtheta to 2pi d varphi(z_n).
    $$
    The integrand in the second term is a continuous function and thus
    $$
    Big|int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)dsBig| leq sup |varphi|cdot sup|partial_r log |G||cdot 2piepsilon to 0.
    $$
    Summing up over all $n$, we see that
    $$
    lim_{epsilonto 0}int_{U_varepsilon} fDeltavarphi dx = sum_n 2pi d_n varphi(z_n),
    $$
    as desired.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      About the term $int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds$, note that $partial U_epsilon = bigcup_n partial B(z_n,epsilon)$. Fix $n$ and observe that
      $$
      F(z) = (z-z_n)^d G(z)
      $$
      where $G(z)$ does not vanish near $z=z_n$. This implies
      $$
      f(z) = dlog |z-z_n| + log|G(z)|=dlog epsilon + log|G(z)|
      $$
      on $|z-z_n|=epsilon$. Note that $log |G(z)|$ is a continuous harmonic function and hence $|log |G(z)||leq C$ for some $C>0$. Now, we see that
      $$
      int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =dlogepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle ds +int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle ds.
      $$
      From
      $$
      dBig|logepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle dsBig| = d logepsilon^{-1} int_{|z-z_n|=epsilon} |nabla varphi| dsleq d logepsilon^{-1} cdot 2piepsilon cdotsup|nabla varphi|to 0,
      $$
      and
      $$
      Big|int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle dsBig|leq 2pi Cepsilon sup|nablavarphi|to 0,
      $$
      we get $$
      lim_{epsilonto 0}int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =0.
      $$
      This holds for all $n$, and we conclude $lim_{epsilonto 0}int_{partial U_epsilon} ffrac{partialvarphi}{partial n}ds=0.$



      Next, note that $$frac{partial f(z_n + re^{itheta})}{partial n} = frac{partial}{partial r} f(z_n + re^{itheta})= frac{partial}{partial r}[dlog r + log |G(z_n+re^{itheta})|] = frac{d}{r}+partial_rlog |G(z_n+re^{itheta})|. $$ This is because the unit outward normal vector $n$ of the circle $|z-z_n|=r$ at $z = z_n +re^{itheta}$ is just $e^{itheta}$. Now observe that
      $$
      int_{|z-z_n|=epsilon} varphifrac{partial f}{partial n}ds = frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds + int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)ds .
      $$
      The first term is
      $$
      frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds= frac{d}{epsilon}int_0^{2pi} varphi(z_n + epsilon e^{itheta} )epsilon dtheta =dint_0^{2pi} varphi(z_n + epsilon e^{itheta} )dtheta to 2pi d varphi(z_n).
      $$
      The integrand in the second term is a continuous function and thus
      $$
      Big|int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)dsBig| leq sup |varphi|cdot sup|partial_r log |G||cdot 2piepsilon to 0.
      $$
      Summing up over all $n$, we see that
      $$
      lim_{epsilonto 0}int_{U_varepsilon} fDeltavarphi dx = sum_n 2pi d_n varphi(z_n),
      $$
      as desired.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        About the term $int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds$, note that $partial U_epsilon = bigcup_n partial B(z_n,epsilon)$. Fix $n$ and observe that
        $$
        F(z) = (z-z_n)^d G(z)
        $$
        where $G(z)$ does not vanish near $z=z_n$. This implies
        $$
        f(z) = dlog |z-z_n| + log|G(z)|=dlog epsilon + log|G(z)|
        $$
        on $|z-z_n|=epsilon$. Note that $log |G(z)|$ is a continuous harmonic function and hence $|log |G(z)||leq C$ for some $C>0$. Now, we see that
        $$
        int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =dlogepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle ds +int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle ds.
        $$
        From
        $$
        dBig|logepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle dsBig| = d logepsilon^{-1} int_{|z-z_n|=epsilon} |nabla varphi| dsleq d logepsilon^{-1} cdot 2piepsilon cdotsup|nabla varphi|to 0,
        $$
        and
        $$
        Big|int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle dsBig|leq 2pi Cepsilon sup|nablavarphi|to 0,
        $$
        we get $$
        lim_{epsilonto 0}int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =0.
        $$
        This holds for all $n$, and we conclude $lim_{epsilonto 0}int_{partial U_epsilon} ffrac{partialvarphi}{partial n}ds=0.$



        Next, note that $$frac{partial f(z_n + re^{itheta})}{partial n} = frac{partial}{partial r} f(z_n + re^{itheta})= frac{partial}{partial r}[dlog r + log |G(z_n+re^{itheta})|] = frac{d}{r}+partial_rlog |G(z_n+re^{itheta})|. $$ This is because the unit outward normal vector $n$ of the circle $|z-z_n|=r$ at $z = z_n +re^{itheta}$ is just $e^{itheta}$. Now observe that
        $$
        int_{|z-z_n|=epsilon} varphifrac{partial f}{partial n}ds = frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds + int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)ds .
        $$
        The first term is
        $$
        frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds= frac{d}{epsilon}int_0^{2pi} varphi(z_n + epsilon e^{itheta} )epsilon dtheta =dint_0^{2pi} varphi(z_n + epsilon e^{itheta} )dtheta to 2pi d varphi(z_n).
        $$
        The integrand in the second term is a continuous function and thus
        $$
        Big|int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)dsBig| leq sup |varphi|cdot sup|partial_r log |G||cdot 2piepsilon to 0.
        $$
        Summing up over all $n$, we see that
        $$
        lim_{epsilonto 0}int_{U_varepsilon} fDeltavarphi dx = sum_n 2pi d_n varphi(z_n),
        $$
        as desired.






        share|cite|improve this answer











        $endgroup$



        About the term $int_{partial U_varepsilon} ffrac{partialvarphi}{partial n}ds$, note that $partial U_epsilon = bigcup_n partial B(z_n,epsilon)$. Fix $n$ and observe that
        $$
        F(z) = (z-z_n)^d G(z)
        $$
        where $G(z)$ does not vanish near $z=z_n$. This implies
        $$
        f(z) = dlog |z-z_n| + log|G(z)|=dlog epsilon + log|G(z)|
        $$
        on $|z-z_n|=epsilon$. Note that $log |G(z)|$ is a continuous harmonic function and hence $|log |G(z)||leq C$ for some $C>0$. Now, we see that
        $$
        int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =dlogepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle ds +int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle ds.
        $$
        From
        $$
        dBig|logepsilon int_{|z-z_n|=epsilon} langle nabla varphi,nrangle dsBig| = d logepsilon^{-1} int_{|z-z_n|=epsilon} |nabla varphi| dsleq d logepsilon^{-1} cdot 2piepsilon cdotsup|nabla varphi|to 0,
        $$
        and
        $$
        Big|int_{|z-z_n|=epsilon} log|G(z)|langle nabla varphi,nrangle dsBig|leq 2pi Cepsilon sup|nablavarphi|to 0,
        $$
        we get $$
        lim_{epsilonto 0}int_{|z-z_n|=epsilon} ffrac{partialvarphi}{partial n}ds =0.
        $$
        This holds for all $n$, and we conclude $lim_{epsilonto 0}int_{partial U_epsilon} ffrac{partialvarphi}{partial n}ds=0.$



        Next, note that $$frac{partial f(z_n + re^{itheta})}{partial n} = frac{partial}{partial r} f(z_n + re^{itheta})= frac{partial}{partial r}[dlog r + log |G(z_n+re^{itheta})|] = frac{d}{r}+partial_rlog |G(z_n+re^{itheta})|. $$ This is because the unit outward normal vector $n$ of the circle $|z-z_n|=r$ at $z = z_n +re^{itheta}$ is just $e^{itheta}$. Now observe that
        $$
        int_{|z-z_n|=epsilon} varphifrac{partial f}{partial n}ds = frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds + int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)ds .
        $$
        The first term is
        $$
        frac{d}{epsilon}int_{|z-z_n|=epsilon} varphi ds= frac{d}{epsilon}int_0^{2pi} varphi(z_n + epsilon e^{itheta} )epsilon dtheta =dint_0^{2pi} varphi(z_n + epsilon e^{itheta} )dtheta to 2pi d varphi(z_n).
        $$
        The integrand in the second term is a continuous function and thus
        $$
        Big|int_{|z-z_n|=epsilon} varphi left(partial_r|_{r=epsilon}log |G(z_n +re^{itheta})|right)dsBig| leq sup |varphi|cdot sup|partial_r log |G||cdot 2piepsilon to 0.
        $$
        Summing up over all $n$, we see that
        $$
        lim_{epsilonto 0}int_{U_varepsilon} fDeltavarphi dx = sum_n 2pi d_n varphi(z_n),
        $$
        as desired.







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        edited Dec 31 '18 at 14:42

























        answered Dec 31 '18 at 14:35









        SongSong

        15.7k1738




        15.7k1738






























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