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# Question about Complex Analysis (power series etc.) watch

1. http://www2.imperial.ac.uk/~bin06/M2...pm3l21(11).pdf

Page 2. Cor. 1. What does this corollary state? Is it saying that if f has a sequence of zeros zn and the set {zn: n is in N} has a limit point in D, then f is constant zero? Also, is a zero just a point where (at least) the first coefficient of the Taylor series about the point is zero? Is the proof of this corollary immediate or has the proof been omitted here? I don't see how it comes about.

Thanks for any help.
2. being a zero means that ; as such the corollary states that if f is holomorphic in D and with all zeroes of f, then f is identically zero in D (i.e. every point of D is a zero of f).
3. (Original post by nuodai)
being a zero means that ; as such the corollary states that if f is holomorphic in D and with all zeroes of f, then f is identically zero in D (i.e. every point of D is a zero of f).

I see. So what is so special about this result? Does it just basically tell us that if f is zero at countably infinite many points then it is zero everywhere in its domain?
4. (Original post by gangsta316)
I see. So what is so special about this result? Does it just basically tell us that if f is zero at countably infinite many points then it is zero everywhere in its domain?
You mean a dense set, not a countably infinite set. If not sin and cos would be identically zero.

It is important as it gives you the identity theorem which is used in analytic continuation, finding a holomorphic function with a larger domain but that agrees on the original one.
5. (Original post by gangsta316)
I see. So what is so special about this result? Does it just basically tell us that if f is zero at countably infinite many points then it is zero everywhere in its domain?
On top of what thebadgeroverlord said, it's useful because it can tell you that some functions aren't analytic in certain domains. Consider the function in some region about the origin for example; it has zeroes at and clearly which lies in the domain. If this function were to be analytic, it would be identically zero, but it isn't, so it can't be analytic.

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