A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch9.2

I have questions on the proof of Casorati-Weierstrass Theorem (Thm 9.7)

-

If $z_0$ is an essential singularity of $f$ and $r$ is any positive real number, then every $w \in \mathbb C$ is arbitrarily close to a point in $f(\{0<|z-z_0|<r\})$. That is, for any $w \in \mathbb C$ and any $\varepsilon > 0$ there exists $z \in \{0<|z-z_0|<r\}$ such that $|w − f(z)| < \varepsilon$.

-

Note: The book uses $D\!\!\!\cdot$ $[z_0,r]$ to denote a punctured disc $\{0<|z-z_0|<r\}$.

-

enter image description here

-

  1. ($\color{blue}{\text{blue}}$ box) Why do we have $$\lim_{z \to z_0} \frac{z-z_0}{f(z)-w} = 0$$? I tried:
  • Pf (1): Let $\varepsilon > 0,$ we must find $\delta > 0$ s.t. $$|\frac{z-z_0}{f(z)-w}| < \varepsilon \ \text{whenever} \ |\frac{z-z_0}{1}| < \delta.$$

  • We're given that there exists $M, \delta_1 > 0$ s.t. $$|\frac{1}{f(z)-w}| \le M \ \text{whenever} \ |\frac{z-z_0}{1}| < \delta_1$$

Therefore, for all $\varepsilon > 0$, by choosing $\delta := \min\{\frac{\varepsilon}{M},\delta_1\}$, we have

$$|\frac{z-z_0}{f(z)-w}| \le |\frac{z-z_0}{1}\frac{1}{f(z)-w}|$$

$$ \le |\frac{z-z_0}{1}|M < \varepsilon \ \text{whenever} \ |\frac{z-z_0}{1}| < \delta$$

QED

  1. ($\color{green}{\text{green}}$ box) That $z_0$ a singularity of $g$: Where is this used in the rest of the proof?

I suspect none and hence the round brackets.

  1. (1st $\color{red}{\text{red}}$ box) Why is $z_0$ a singularity of $g$?

I understand that if $z_0$ is a singularity, $z_0$ is isolated and removable. I just want to know why $z_0$ is a singularity of $g$ in the first place.

  1. (2nd $\color{red}{\text{red}}$ box) Why is $z_0$ a singularity of $f(z)-w$?

I (think I) understand that if $z_0$ is a singularity, $z_0$ is isolated and then either removable or pole. I just want to know why $z_0$ is a singularity of $f(z)-w$ in the first place.

  1. ($\color{yellow}{\text{yellow}}$ box) Why is $z_0$ a removable singularity of $f(z)-w$ if $z_0$ is not a pole of $f(z)-w$? I tried:

Pf (5): By Prop 9.5b (*), $z_0$ is removable or for all $p \in \mathbb N$, $$\lim (z-z_0)^{p+1}(f(z)-w) \ne 0$$ However, for $p=n-1$, we have $$\lim (z-z_0)^{p+1}(f(z)-w)=0$$ Therefore, $z_0$ is removable. QED (5)

  1. ($\color{black}{\text{black}}$ box) Why is $z_0$ a pole of $f(z)-w$ if $z_0$ is not a removable singularity of $f(z)-w$? I tried:

Pf (6): Observe that $$\lim (z-z_0)^n(f(z)-w) = 0$$

By assumption, $z_0$ is not removable. Therefore, by Prop 9.5b (*), $z_0$ is a pole of order $n-1$. QED (6)


(*) (Prop 9.5) Suppose $z_0$ is an isolated singularity of $f$. Then

(a) $z_0$ is removable iff $\lim_{z \to z_0} (z-z_0) f(z) = 0$

(b) $z_0$ is a pole iff $z_0$ is not removable and $\exists n \in \mathbb N: \lim_{z \to z_0} (z-z_0)^{n+1} f(z) = 0$

put on hold as unclear what you're asking by Did, Xander Henderson, amWhy, max_zorn, Parcly Taxel Aug 16 at 4:33

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • the question does not show up – onurcanbektas Aug 14 at 7:23
  • @onurcanbektas what do you mean by show up please? – BCLC Aug 14 at 8:22
  • The first quote box is empty. – onurcanbektas Aug 14 at 10:40
  • 1
    @onurcanbektas it's supposed to say: Why do we have $$\lim_{z \to z_0} \frac{z-z_0}{f(z)-w} = 0$$? – BCLC Aug 14 at 10:46
  • 1
    Thanks for re-quote, my internet might have some problems. – onurcanbektas Aug 14 at 10:59
up vote 1 down vote accepted
+50

1)bounded $\times$ infinitesimal = infinitesimal

2)Not used eslewhere; just an additional info

3)-4)because if $z_0$ is a singularity for $f$ then it is such also for composition with $f$ like $\frac{z-z_0}{f(z)-w}$ or $f(z)-w$.

5-6)if it is clear that $f(z)-w$ has a singularity at $z_0$, then you know a singularity can be of three types; since $\lim_{z\to z_0}(z-z_0)^n(f(z)-w)=0$ this mean that, if $n=0$ then $f$ has a removable singularity at $z_0$; if $n\ge1$ what's happen?

  • Thanks Joe! Good intuition re 1 – BCLC Aug 13 at 9:46

Let's consider the proof step by step

  1. Please avoid writing implication in the converse order: it's always better to specify the assumption first and then proceed with the proof, for the sake of readability. Nonetheless, the proof is correct.

  2. Indeed, nowhere. It's just an info.

  3. Because if $f$ has a singularity at $z_0$ then $g\circ f$ must also have one there.
  4. An isolated singularity is an isolated point where the function is not defined.
  5. and 6. Yes. Indeed removable is so to say a "pole of order zero", hence just a special case.
  • Thanks b00n heT! Good intuition re pole of order zero. – BCLC Aug 13 at 9:46

Not the answer you're looking for? Browse other questions tagged or ask your own question.