The first time when I see Casorati-Weierstrass' Theorem in Complex Analysis,I was shocked:

Casorati-Weierstrass' Theorem suppose $f$ is holomorphic in the punctured disc $D_r(z_0)-\{z_0\}$ and has an essential singularity at $z_0$. Then,the image of $D_r(z_0)-\{z_0\}$ under $f$ is dense in the complex plane.

How did Casorati and Weierstrass find this theorem? Did they explore some specific example and find this phenomenon? There is a classical complex function to give an example for essential singularity in many books. It is $$f(z)=\exp\left(\frac{1}{z}\right)$$

At $z_0=0$,it is not a removable singularity and Pole singularity.so it is a essential singularity. And I want to see the above phenomenon($f$ is dense in the complex plane) from this special instance. Set $z=re^{i\theta}$,then $$f(z)=\exp\left(\frac{\cos\theta}{r}\right)\cos\left(\frac{\sin\theta}{r}\right)-i\exp\left(\frac{\cos\theta}{r}\right)\sin\left(\frac{\sin\theta}{r}\right)$$ If I consider $f$ as a mapping from $Z$ to $W$. Did they find this phenomenon from studying the image in complex plane $W$? For simplicity, I consider the image of circle in $Z$ plane. For a fixed $r$,and $0\leq\theta <2\pi$,the below parametric equations give a image in $W$ plane: $$\begin{cases}x(r,\theta)=\exp\left(\frac{\cos\theta}{r}\right)\cos\left(\frac{\sin\theta}{r}\right)\\y(r,\theta)=\exp\left(\frac{\cos\theta}{r}\right)\sin\left(\frac{\sin\theta}{r}\right) \end{cases}$$

For a large $r$, the image looks like a point in real axis with coordinate$(1,0)$.so the reason cause its dense in $W$ must be ascirbed to $r$ near $0$. then I want to see the exact image shape in $W$. then I used Maxima to draw the figure for $r$ near $1$.: enter image description here

It is beautiful. and axial symmetry of real-axis. But I can't see some deep phenomenon. How can we find this phenomenon from study this special instance?


1 Answer 1


It is a spectacular theorem, and the full truth is even more spectacular. If $f$ has an essential singularity at $z=z_0$, then the equation $f(z) = w$ will have infinitely many solutions in each (punctured) neighbourhood of $z_0$, with at most one exceptional value (of $w$). This is known as Picard's big theorem.

To verify Weierstrass-Casorati (or even big Picard) for this particular example, you can just solve the equation

$$\exp\big(\frac1z\big) = w$$

whose solutions are

$$z = \frac{1}{\log w} = \frac{1}{\ln|w| + i\operatorname{Arg} w + 2\pi i k}$$

for $k \in \mathbb{Z}$. It's easy to see that for every $w\neq 0$, there are solutions $z$ whose modulus is arbitrarily small (by choosing a large value of $k$).


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