# Veryfiy Picard's Theorem

## 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
Homework Helper
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.

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

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.