Veryfiy Picard's Theorem

[usn7564]

Homework Statement

Verify Picard's Theorem for cos(1/z) at z = 0

Homework Equations

The theorem:

A function with an essential singularity assumes every complex number, with possibly one exception, as a value in any neighborhood of this singularity

The Attempt at a Solution

I have a solution that I don't understand (from a manual). What the manual does is break out z to get
$$z = \frac{1}{log(c\pm \sqrt{c^2-1}}$$ for an arbitrary complex constant c. Getting there is trivial enough but I don't get the conclusion.

Values of the logarithm can be chosen to make $$|z| < \epsilon$$ for any positive $$\epsilon$$, so that cos(1/z) achieves the value c in any neighborhood of z = 0.

I understand that you can make z arbitrarily small by choosing a k high enough in the logarithm, but intuitively I'd think cos(z) wouldn't leave [-1, 1] no matter how fast it jumped between them. Intuition be damned, I can't see the reasoning at all. Because I can get a very large input in the cosine function I know the function can output any complex number? How?

Guessing it's something rather silly I'm missing here but there you go.
Any help would be appreciated, cheers.

 

[Physics Monkey]

I think you're missing the fact that $\cos{w}$ does not necessarily sit in the interval $[-1,1]$ when $w$ is complex. For example, if $w$ is pure imaginary, then $\cos{w} = \cosh{|w|}$.

So the point is that you can use the unboundedness from a large imaginary part of $w =1/z$ plus the phase degree of freedom from the real part of $w$ to get more or less any complex number you want.

 

[usn7564]

You're right, that completely flew me by. Thanks, wasn't that tricky after all.

 

[jackmell]

How?

First, better define what you mean by Picard's Theorem. He has two, the little one and the big one.

Little: Every entire function that's not a polynomial has an essential singularity at infinity.

Big: A function with an essential singularity achieves every value, with at most one exception, infinitely often in any neighborhood of the singularity.

I think you want the big one.

Any help would be appreciated, cheers.

First do a Wikipedia on Casorati-Weierstrass, study the example for $e^{1/z}$, then apply that example to your problem.