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How can I verify Casorati-Weierstrass theorem on the example ?$$f(z)=\sin\frac{1}{z}$$

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I think you meant to link to a different page. –  Michael Albanese Jan 9 '13 at 6:16
    
@DisplayName do you want check that every ball centerd in $0$ has image dense in $\mathbb C$? –  user52188 Jan 9 '13 at 6:19
    
that I did ^^ math.stackexchange.com/questions/154703/… –  Display Name Jan 9 '13 at 6:19
    
@DisplayName or do you want prove that in $0$ f has a essential singularity? for then you apply Casorati-Weierstrass? –  user52188 Jan 9 '13 at 6:23
    
@EdgarMatias The problem is not worded that specific. That's why I can only guess. I think, I probably have to show that f meets the requirements for Casorati-Weierstrass(the essential singularity) and that its dense in 0. I dont think I can simply apply the theorem –  Display Name Jan 9 '13 at 7:32

1 Answer 1

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You need to verify the Laurent series expansion has infinitely negative terms. But this should not be difficult because you can compare the number $$\frac{e^{\frac{i}{z}}-e^{-\frac{i}{z}}}{2i}$$with arbitrally given $\frac{1}{z^{n}}$ near 0. Or maybe you can expand straightforwardly. I do not know what else you need to "verify" (big or small) Picard's theorem in this case.

To visualize go to

http://www.wolframalpha.com/input/?i=Plot+sin[1%2Fz]

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Do you maybe mean like this: $$sin\frac{1}{z}=\sum_{n=-1}^{-infinity} \frac{(-1)^n z^{2n+1}}{(2n+1)!}$$ –  Display Name Jan 9 '13 at 8:10
    
Yes. That's what I mean. –  Bombyx mori Jan 9 '13 at 8:13
    
This is almost certainly not the intension of the exercise. –  mrf Jan 9 '13 at 9:31

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