# holomorphic function convergent sequence identically zero Watch

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Hi

Theorem attached and proof.

I am stuck on:

1) Where we get comes from so is the first non-zero fourier coeffient. So I think this term is , from the radius of the open set, but I don't know how to take care of the rest of the higher tems through , is this some theorem or?

2) The conclusion thus has only one zero at I think i'm being stupid but what is this being made from? We know and , but I dont understand.

Thanks

Theorem attached and proof.

I am stuck on:

1) Where we get comes from so is the first non-zero fourier coeffient. So I think this term is , from the radius of the open set, but I don't know how to take care of the rest of the higher tems through , is this some theorem or?

2) The conclusion thus has only one zero at I think i'm being stupid but what is this being made from? We know and , but I dont understand.

Thanks

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#4

For (1), it's literally just continuity. g is cts and |g(z_0)| > 0. By cty of g,, we can find R s.t.. |z-z0| < R => |g(z) - g(z_0)| < |g(z_0)|/2. Shrink r so that |z-z0| <R is always satisfied and you're done.

For (2), well, if we can write f as (z-z0)^m g(z), and g is never zero near z0, how can f be 0 unless z = z0?

For (2), well, if we can write f as (z-z0)^m g(z), and g is never zero near z0, how can f be 0 unless z = z0?

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(Original post by

and g is never zero near z0,

**DFranklin**)and g is never zero near z0,

I don't understand how implies this though

This choice seems a bit random to be, I dont understand it's significance.

Thanks

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#6

(Original post by

Thanks

I don't understand how implies this though

This choice seems a bit random to be, I dont understand it's significance.

Thanks

**xfootiecrazeesarax**)Thanks

I don't understand how implies this though

This choice seems a bit random to be, I dont understand it's significance.

Thanks

**at**z0, and because it's cts, we can find a neighbourhood of z0 where g(z) is always close to g(z0). Then since g(z0) isn't 0, g(z) isn't either.

This is a basic continuity argument.

Can I ask what your background is, because it seems you're attempting a postgraduate course in pure mathematics with gaps in your knowledge that should have been covered in the first year of an undergraduate degree.

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(Original post by

g is never zero near z0 because it's non-zero

This is a basic continuity argument.

Can I ask what your background is, because it seems you're attempting a postgraduate course in pure mathematics with gaps in your knowledge that should have been covered in the first year of an undergraduate degree.

**DFranklin**)g is never zero near z0 because it's non-zero

**at**z0, and because it's cts, we can find a neighbourhood of z0 where g(z) is always close to g(z0). Then since g(z0) isn't 0, g(z) isn't either.This is a basic continuity argument.

Can I ask what your background is, because it seems you're attempting a postgraduate course in pure mathematics with gaps in your knowledge that should have been covered in the first year of an undergraduate degree.

doing masters in math

math and physics undergrad.

the course listed background material as complex analysis only

but yes it seems to have links to pure/algebra which were not listed as prerequisites and I did very little pure during my undergrad.

*in fact if done in first year I've most likely forgotten some

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#8

(Original post by

ta

doing masters in math

math and physics undergrad.

the course listed background material as complex analysis only

**xfootiecrazeesarax**)ta

doing masters in math

math and physics undergrad.

the course listed background material as complex analysis only

Certainly this would have been regarded as a straightforward argument in my complex analysis course.

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(Original post by

Thing is, I would assume real analysis as a prerequisite for complex analysis (and I'd be very surprised if your university doesn't).

Certainly this would have been regarded as a straightforward argument in my complex analysis course.

**DFranklin**)Thing is, I would assume real analysis as a prerequisite for complex analysis (and I'd be very surprised if your university doesn't).

Certainly this would have been regarded as a straightforward argument in my complex analysis course.

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#10

(Original post by

fair point, I'm pretty sure we didn't use much analysis in the complex analysis course, if so I've forgot it.

**xfootiecrazeesarax**)fair point, I'm pretty sure we didn't use much analysis in the complex analysis course, if so I've forgot it.

At the least, I would read back over your complex variable notes, or if you really didn't do anything on the "analysis" side for complex variable, see if there's a text book your lecturer can recommend.

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#11

(Original post by

For (1), it's literally just continuity. g is cts and |g(z_0)| > 0. By cty of g,, we can find R s.t.. |z-z0| < R => |g(z) - g(z_0)| < |g(z_0)|/2. Shrink r so that |z-z0| <R is always satisfied and you're done.

**DFranklin**)For (1), it's literally just continuity. g is cts and |g(z_0)| > 0. By cty of g,, we can find R s.t.. |z-z0| < R => |g(z) - g(z_0)| < |g(z_0)|/2. Shrink r so that |z-z0| <R is always satisfied and you're done.

Else you don't get the lower bound on g(z)?

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#12

**xfootiecrazeesarax**)

fair point, I'm pretty sure we didn't use much analysis in the complex analysis course, if so I've forgot it.

https://www.amazon.co.uk/Schaums-Out.../dp/0071615695

It's a fairly fast-paced but decent overview of complex analysis, and is presented in the standard Schaum "solved problems" format. It may be a bit light on the hard-core analysis side (e.g. I don't think it mentions the identity theorem, which I think you're using/proving here), but that may be what you need at the moment.

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#13

(Original post by

Maybe I'm confused, but don't you want something like this here:

Else you don't get the lower bound on g(z)?

**atsruser**)Maybe I'm confused, but don't you want something like this here:

Else you don't get the lower bound on g(z)?

[I'm aware I may seem a bit harsh, but having done a post graduate course that was fairly similar to this, I'm aware how it's assumed you can "fill in the details" on things like this (and indeed much more complicated than this)].

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#14

(Original post by

I don't think there's a problem with what I wrote;. You'll need to use |a-b| < C implies |b|>|a|-C but that should be obvious at this level.

**DFranklin**)I don't think there's a problem with what I wrote;. You'll need to use |a-b| < C implies |b|>|a|-C but that should be obvious at this level.

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#15

(Original post by

No, there isn't a problem - it's just that when you said "And you're done", it looked like you'd omitted half the argument, and claimed that it was the finished item. In fact I see that you were only talking about the continuity bit - fair enough, though maybe confusing to the OP.

**atsruser**)No, there isn't a problem - it's just that when you said "And you're done", it looked like you'd omitted half the argument, and claimed that it was the finished item. In fact I see that you were only talking about the continuity bit - fair enough, though maybe confusing to the OP.

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