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gotjrgkr
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Homework Statement
Hello!
I have a question about a meromorphic function introduced in a field of complex analysis.
I refer to palk's book "an introduction to complex function theory".
As you can check in that book, the author introduces a definition of a meromorphic function in p. 318. I quote the definition here; " We characterize a function f as meromorphic in an open set U provided f has at no point of U worse than a pole.". you can also find the expression " a function f has no worse than a pole at a point z[itex]_{0}[/itex]" meaning that there exists a radius r > 0 such that the given function f is either analytic in the full open disk Δ(z[itex]_{o}[/itex],r) or else analytic in the punctured disk Δ[itex]^{\ast}[/itex](z[itex]_{0}[/itex],r) with a pole or removable singularity at its center. You can check this in p. 317 in the book.
Now, if you look at a paragraph right above "2.5 Essential Singularities" in p. 319, it is said that if functions f and g are both meromorphic in an open set U, then you can get directly from the th.2.5 in p.318 f/g are likewise meromorphic in this open set unless g is identically zero in some component of U.
I think I misunderstand something here, so that I can't see the result.
Homework Equations
The Attempt at a Solution
I've tried to apply th. 2.5 in p. 318 to prove the remark witten in p. 319 which I've written right above. I think there would be a problem when g has a removable singularity at z[itex]_{0}[/itex] with the limit limit[itex]_{z\rightarrowz_{0}}[/itex]g(z)=0 where z[itex]_{0}[/itex] is a point of a component of U. What I mean is I can't safely say that g is defined in a punctrued disk around z[itex]_{0}[/itex]. I think I need to gaurantee (from the assumption g is not identically zero in any component of U) that for any point in a component of the open set U which is a removable singular point with limit[itex]_{z\rightarrowz_{0}}[/itex]g(z)=0, there's a punctured disk around that point where f is not identically zero, so that I can apply corollary 1.3 in p. 302 to say safely that f/g is well defined in a punctured disk centered at z[itex]_{0}[/itex].
But, I can't prove it. Furthermore, I expect there's a example such that under the assumption g is not identically zero in any component of U, f/g can't be well defined.
; Let U be an open disk centered at 0, g be a function whose domain set is the disk and its value is one at zero and zero otherwise. (f is any analytic function in the disk)
Is there anyone who can explain about this matter?