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usn7564

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## 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

[tex]z = \frac{1}{log(c\pm \sqrt{c^2-1}}[/tex] 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 [tex] |z| < \epsilon[/tex] for any positive [tex] \epsilon[/tex], 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.