Complex analysis- poles vs. Zeros, etc.

quasar_4
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I am having a hard time understanding the difference between poles and zeros, and simple poles versus removable poles. For instance, consider f(z)=\frac{z^2}{sin(z)}. we can expand sine into a power series and pull out a z, so doesn't that remove the singularity at z=0? Also, I don't see why n*pi would not also be removable since it doesn't seem to be a problem in the series expansion (but according to my graded homework, 0 is a zero and n*pi is a simple pole)... Can someone help me out here?
 
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quasar_4 said:
we can expand sine into a power series and pull out a z, so doesn't that remove the singularity at z=0?
That's how you remove the singularity. But this operation produces a new (partial) function that is not f. (The difference being that this function is defined at 0 whereas f is not)
 
But 0 is not a zero, it is a removable singularity! :/
 
quasar987 said:
But 0 is not a zero, it is a removable singularity! :/
?? What is your point? Hurkyl's point was that if f(z) has a "removable singularity" at z_0, yes, you can "remove" it but then you get a different function, g(z). g(z)= f(z) for all z except z_0. He never said anything about being a zero.
 
My comment was in response to
quasar_4 said:
(but according to my graded homework, 0 is a zero and n*pi is a simple pole)... Can someone help me out here?
HallofIvy.
 
To try and sum up:
1) Cancel z top and bottom to show that the bottom term -> 1 as z -> 0. So that would remove the singularity and make the function analytic at zero.
1a) Because the bottom can't go to zero, the function must -> 0 when z -> 0. So there is a zero of the function at z = 0.
2) But if you don't cancel the z and stick with the original function, the sin(z) will vanish every time z -> n*pi and the function will go through the roof. So there are simple poles when z = n*pi.

hope this helps.
 
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