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Veryfiy Picard's Theorem

  • Thread starter usn7564
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  • #1
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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.
 

Answers and Replies

  • #2
Physics Monkey
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I think you're missing the fact that [itex] \cos{w}[/itex] does not necessarily sit in the interval [itex] [-1,1] [/itex] when [itex] w [/itex] is complex. For example, if [itex] w [/itex] is pure imaginary, then [itex] \cos{w} = \cosh{|w|} [/itex].

So the point is that you can use the unboundedness from a large imaginary part of [itex] w =1/z[/itex] plus the phase degree of freedom from the real part of [itex] w[/itex] to get more or less any complex number you want.
 
  • #3
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You're right, that completely flew me by. Thanks, wasn't that tricky after all.
 
  • #4
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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 [itex]e^{1/z}[/itex], then apply that example to your problem.
 

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