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Let $f(z)$ be analytic in the disk $|z|<R \ \ \ (R>1)$ except for a simple pole at a point $z_0$, $|z_0|=1$. Consider the expansion $f(z)=a_0+a_1 z+ \cdots$, and show that $$\lim_{n \to \infty} \frac {a_n} {a_{n+1}}=z_0$$

All my attempts failed. I wanted to use the Laurent series at $z_0$ but the problem needs expansion at $0$.

Siminore
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Gobi
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2 Answers2

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There is an $a\ne0$ such that the principal part of $f$ at $z_0$ is given by $z\mapsto{a\over z-z_0}$. The function $g(z):=f(z)-{a\over z-z_0}$ has a removable singularity at $z_0$ and is analytic otherwise in $D_R$, which implies that $g$ is in fact analytic in $D_R$. So $g$ has a Taylor expansion $g(z)=\sum_{n=0}^\infty b_nz^n$ which is convergent for $|z|<R$, in particular at $z:={1\over2}(1+R)$. Therefore one necessarily has $$|b_n|\leq\Bigl({2\over1+R}\Bigr)^n\qquad(n>n_0)\ ;$$ which implies $\lim_{n\to\infty} b_n=0$. On the other hand, $${a\over z-z_0}=-{a/z_0 \over 1-(z/z_0)}=-{a\over z_0}\sum_{n=0}^\infty\Bigl({z\over z_0}\Bigr)^n\ .$$ Now $f(z)={a\over z-z_0}+g(z)$. Therefore the Taylor coefficients $a_n$ of $f$ are given by $$a_n=-{a\over z_0^{n+1}} + b_n\ ,$$ from which we get $${a_n\over a_{n+1}}={-a/z_0^{n+1} +b_n\over -a/z_0^{n+2}+b_{n+1}}=z_0{a-b_n z_0^{n+1} \over a - b_{n+1}z_0^{n+2}}\to z_0\qquad(n\to\infty)\ .$$

Christian Blatter
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  • Thanks, I understood all the things but one: In first sentence, how can you notice instantly the existence of $a$ and $g(z)$? My understanding lacks something: Since $f(z)$ has a pole at $z_0$, it can be expressed as Laurent series at $z_0$, so $f(z)= a/(z-z_0) + g(z)$ where $g(z)$ is analytic inside the disk of radius $R-1$ at $z_0$. So the analytic radius of $g(z)$ is different for yours. – Gobi Sep 17 '12 at 13:14
  • @Gobi: I have explained it in my edit. – Christian Blatter Sep 17 '12 at 13:46
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Consider $\displaystyle f(z) = \frac{g(z)}{z-z_0}$. Since $f$ has a simple pole at $z_0$, $g$ is holomorphic on the disc.

Now, let $f = \sum a_nz^n, g=\sum b_nz^n$. Then, $\sum b_nz^n = (z-z_0)\sum a_nz^n$. Comparing the coefficients, you get $b_{n+1} = a_n - z_0a_{n+1}$.

As $n \to \infty$, $b_n \to 0$ while $a_n$ will not tend to $0$.

aviness
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  • I appreciate your answer, but how can I get the last statement? – Gobi Sep 15 '12 at 01:16
  • Well, $b_n \to 0$ as a result of the radius of convergence being $>1$. However, we need something a bit stronger than $a_n \not\to 0$ to conclude the claim. For instance if $a_n=0$ infinitely often the limit is not even well-defined. I think to make this argument work we need to $a_n$ bound away from $0$ for all large $n$. – Erick Wong Sep 15 '12 at 03:30