# [SOLVED] Prove that an essential singularity approaches every complex number.

• Mar 12th 2010, 03:42 PM
davismj
[SOLVED] Prove that an essential singularity approaches every complex number.
http://i39.tinypic.com/2j5hxqo.jpg

I'm not sure if they're suggesting that for any complex number w, a function f with an essential singularity at $\displaystyle z_0$ has the property that $\displaystyle f \to w$ as z $\displaystyle \to z_0$. This seems intuitively to just be not true at all, but to be fair, the book's description of essential singularities is "we leave essential singularities to the exercises."
• Mar 12th 2010, 03:48 PM
Drexel28
Quote:

Originally Posted by davismj
http://i39.tinypic.com/2j5hxqo.jpg

I'm not sure if they're suggesting that for any complex number w, a function f with an essential singularity at $\displaystyle z_0$ has the property that $\displaystyle f \to w$ as z $\displaystyle \to z_0$. This seems intuitively to just be not true at all, but to be fair, the book's description of essential singularities is "we leave essential singularities to the exercises."

Well, which definition did they give you in the exercises. That it's a singularity which is neither removable or a pole or that neither $\displaystyle \lim_{z\to z_0}f(z)$ or $\displaystyle \lim_{z\to z_0}\frac{1}{f(z)}$ exists?
• Mar 12th 2010, 04:11 PM
davismj
Quote:

Originally Posted by Drexel28
Well, which definition did they give you in the exercises. That it's a singularity which is neither removable or a pole or that neither $\displaystyle \lim_{z\to z_0}f(z)$ or $\displaystyle \lim_{z\to z_0}\frac{1}{f(z)}$ exists?

Neither pole nor removable.

They also give the hint: if g is bounded, then show that f has a pole or a removable singularity at $\displaystyle z_0$. This implies that $\displaystyle |f(z) - w| \ne 0$ as $\displaystyle z \to z_0$?