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

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March 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 $z_0$ has the property that $f \to w$ as z $\to z_0$ . 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."
March 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 $z_0$ has the property that $f \to w$ as z $\to z_0$ . 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 $\lim_{z\to z_0}f(z)$ or $\lim_{z\to z_0}\frac{1}{f(z)}$ exists?
March 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 $\lim_{z\to z_0}f(z)$ or $\lim_{z\to z_0}\frac{1}{f(z)}$ 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 $z_0$ . This implies that $|f(z) - w| \ne 0$ as $z \to z_0$ ?