What Is Mary Boas' Theorem III About Analytic Functions and Taylor Series?

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Discussion Overview

The discussion centers around Mary Boas' Theorem III regarding analytic functions and their Taylor series expansions. Participants explore the implications of a function being analytic in a region, the nature of its derivatives, and the conditions under which these derivatives exist and are defined.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the implications of a function being analytic and the existence of higher-order derivatives, questioning whether analyticity can be determined solely by the first derivative.
  • Others highlight the unique properties of complex differentiation, suggesting that higher-order derivatives exist even if not immediately obvious from the first derivative.
  • One participant presents an example using the function z², discussing its derivatives and questioning how it fits into the definition of analyticity.
  • Another participant emphasizes that polynomials, such as z², are considered analytic functions and that having a derivative of zero is still valid within this context.
  • Clarifications are made regarding the notation used for derivatives, with some participants correcting the use of d/dx to d/dz for complex functions.
  • There is a discussion about the conditions under which a function may not be analytic, particularly when derivatives lead to undefined results in certain regions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the clarity of the definition of analyticity and the implications of higher-order derivatives. Confusion remains regarding the relationship between derivatives and analyticity, indicating multiple competing views.

Contextual Notes

Some participants express uncertainty about the definitions and implications of analyticity, particularly in relation to specific examples and the behavior of derivatives in certain regions.

DrPapper
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On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series.

I get the part about a Taylor series, that's pretty straightforward. But what does she mean about this region and the derivatives. I get that there are higher order derivatives. But I wish she would have given an example of one such function. Also, isn't analytic here only tested for by taking the first derivative? So if we do that we can't just assume that there will be higher order derivatives up to the nth order.
 
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DrPapper said:
Isn't analytic here only tested for by taking the first derivative? So if we do that we can't just assume that there will be higher order derivatives up to the nth order.

Ah but that is precisely the great miracle of complex differentiation ! I do not have that book but I am surprised she doesn't introduce this before. And if she doesn't, the first few pages of any introductory book on complex analysis should cover that - or have a look at https://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions
 
wabbit said:
Ah but that is precisely the great miracle of complex differentiation ! I do not have that book but I am surprised she doesn't introduce this before. And if she doesn't, the first few pages of any introductory book on complex analysis should cover that - or have a look at https://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions
I'm still confused. Say for example you have d/dx z^2 which gives by the equivalent power rule 2z, I assume the next derivative gives 2, then next is 0. So how does this satisfy the analytic for all higher derivatives theorem here? Thanks for your time and help by the way. :D
 
I am not sure I get your question here, it sounds as if you're saying that the higher derivatives of z2 are not immediately obvious to you. Can you clarify ?

Because if that's the case, then you're not ready to take a course in complex analysis at all. What is your background in maths, what courses have you completed ?

However if you were able to study and understand the first 670 pages of that book with no issue, then there's obviously something I'm missing here... So I'll stop speculating and will wait for your clarification:)
 
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LOL yes there's been a misunderstanding. I've done multvar calc, ODEs and a few other things. I'm saying that since

d/dx z2 = 2z
d2/dx2=2
d3/dx3=0=dn>3/dn>3

And z2 is analytic then do we continue to call it analytic even though any derivative higher than the third is still just 0? If that's the case, what makes it "not" have a derivative and is not analytic? Is it simply when the derivative is such that plugging in for a value in the region that this complex function is being analyzed leads to an undefined result?

At first I was thinking well if it's going to have a derivative by this definition we must be talking about functions like Sinz or ez but then there's the polynomial one we've discussed and it qualifies under this definition. So I'm uncertain if I get what the definition is telling me. I think perhaps it's the fact that some derivitives "explode" for certain values plugged into them, such as when your denominator gets turned into zero upon plugging in for values in a region. Is this correct?

BTW, your last response made me laugh man. LOL
 
OK then -
(just one detail it bugs me to see "d/dx(z2)", this should be d/dz)
Yes z2, or any polynomial for that matter, is analytic - no proof really needed, it's just an immediate application of the definition. These are in a way the "most analytic" functions one can think of. I don't understand why you think 0 is not a valid derivative, but it is.
 
wabbit said:
OK then -
(just one detail it bugs me to see "d/dx(z2)", this should be d/dz)
Yes z2, or any polynomial for that matter, is analytic - no proof really needed, it's just an immediate application of the definition. These are in a way the "most analytic" functions one can think of. I don't understand why you think 0 is not a valid derivative, but it is.
I wasn't saying it's not valid, though I see how I kinda did say it. LOL But I talked with a math friend and got the issue figured out. It seems it's an issue of the region that is in question. :D Thank you so much for your help though. It was a tough one to explain what my confusion was. We would have been chatting for weeks. LOL Again thank you though. :D
 
LOL I also just noticed what you were saying about the d/dx thing. Yeah, my mistake. I will write d/dz z2 a hundred times tonight as my punishment. I'm silly sometimes:rolleyes:.
 

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