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Analycity of a function

  1. Jul 16, 2006 #1

    Jex

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    Okay I have that a function is analytic at point a if it can be represented by a power series in x-a with a positive radius of convergence.

    Does anyone else have a better way of putting this?

    Thanks
     
  2. jcsd
  3. Jul 17, 2006 #2
    Let [tex] \Omega\subseteq\mathbb{C} [/tex] be an open connected set. Then [tex] f:\Omega\to\mathbb{C}[/tex] is analytic in [tex] \Omega [/tex], if and only if [tex] f[/tex] is holomorphic in [tex] \Omega [/tex].
     
    Last edited: Jul 17, 2006
  4. Jul 17, 2006 #3

    HallsofIvy

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    Iriss, what does it mean to say a function is "holomorf" ("holomorphic"?) in a set. My understanding of "holomorphic" is that it is a synonym for "analytic" so you haven't really answered the question: Was what he gave a good characterization of "analytic" or "holomorphic"?

    Jex, the only change I would make in your definition would be to say
    "is equal to" rather than "can be represented by" since "represented by" is vague.
    The function
    [tex]f(x)= e^{-\frac{1}{x^2}} if x\ne 0, 0 if x= 0[/itex]
    has a power expansion (Taylor's series) about 0 that converges for all x. Is it "represented" by that series?
    Of course, the series converges for all x because it is identically 0! f(x) is not equal to the series anywhere except x= 0.
     
    Last edited: Jul 17, 2006
  5. Jul 17, 2006 #4

    Jex

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    Thanks guys.
     
  6. Jul 17, 2006 #5

    matt grime

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    There are other equivalent definitions, such as the Cauchy Riemann equations, Morera's theorem, and the fact that the real and imaginary parts should be harmonic (which is just a restatement of C-R equations), as well as the fact that it is equivalent to complex differentiability.
     
  7. Jul 17, 2006 #6
    A function is holomorphic in a set if it is holomorphic in every point in the set. The point is that not all holomorphic functions are analytical in any given subset of [tex]\mathbb{C}[/tex].
     
    Last edited: Jul 17, 2006
  8. Jul 17, 2006 #7

    shmoe

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    I don't understand your point, 'tis obvious that the terms "holomorphic" and "analytic" apply to a function on some subset of points, no? Or are you making some distinction between "analytic" and "holomorphic"? They are synonyms in every usage I've ever encountered ("regular" is another synonym).
     
  9. Jul 17, 2006 #8
    I am making a distinction between "analytic" and "holomorphic". Oviously they are they same when in the complex plane, but the same does not apply on [tex]\mathbb{R}[/tex]. Here there are holomorphic functions that are not analytic, the function in HallsofIvy's post being a good example.
     
  10. Jul 17, 2006 #9

    shmoe

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    What definition are you using that considers Hall's example to be "holomorphic" in any way? It's not one I've ever seen before.
     
  11. Jul 17, 2006 #10
    Let [tex]\Omega\subseteq\mathbb{C}[/tex] be an open set and [tex]f:\Omega\to\mathbb{C}[/tex] a function. Then [tex]f[/tex] is holomorphic at a point [tex]z_0\in\Omega[/tex] if for all [tex]z\in\mathbb{C}[/tex] the limit

    [tex]\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}[/tex]

    exists. In the case of
    [tex]f(x)=
    \begin{cases}
    \exp{(-1/x^2)} & x\neq 0\\
    0 & x=0
    \end{cases}
    [/tex]
    since this is defined on [tex]\mathbb{R}[/tex], the limit

    [tex]\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}[/tex]

    exists for all [tex]x\in\mathbb{R}[/tex] and the function is holomorphic/differentiable.
     
    Last edited: Jul 17, 2006
  12. Jul 17, 2006 #11

    shmoe

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    Your definition of a holomorphic function on the reals is the same as it being differentiable (in the real sense) once. I find that an odd thing to do and would like to know what source you are getting this from (if any?).
     
  13. Jul 18, 2006 #12
    Good point. You would like the function to should be differentiable infinitively many times?
     
  14. Jul 18, 2006 #13

    shmoe

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    I would like to know where your definition of holomorphic applied to real functions came from.

    The concept of analytic/holomorphic/regular for complex valued functions just requires one complex derivative to get derivatives of all orders and equality to a power series. The definition of "real analytic" functions captures this consequence-derivatives of all orders and equal to power series as one real derivative is not enough. I can't say I've seen anyone bother defining "real holomorphic" functions, but if they did I can't possibly see why it would differ from "real analytic".
     
  15. Jul 18, 2006 #14
    Well then it is just a question of what we call it. Being picky the definition of a function being holomorphic only apply to complex valued functions. So defining "real holomorphic functions" could be something I just did in the spur of the moment. So the reason I made the distinction between holomorphic and analytic earlier was because in the real case there are (infinitely) differentiable functions that is not analytic.
     
  16. Jul 18, 2006 #15

    matt grime

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    Ok, it could be, but was it? And if so why did you do it?
     
  17. Jul 18, 2006 #16

    shmoe

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    This is an exceptionally bad thing to do, unless you explain that this isn't standard terminology and you're just making it up as you go along.

    This doesn't explain why you would want to distinguish between "holomorphic" and "analytic", they are synonyms.
     
  18. Jul 18, 2006 #17

    HallsofIvy

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    I have seen (don't remember where right now) a function from R to R being defined to be "real analytic" on a set if and only if it is equal to it's Taylor's series on that set, (Which of course requires that it be infinitely differentiable in order that the series exist. The point of my example before was that a function may be infinitely differentiable (on the real numbers) without being "analytic".)

    Iriss, you seem to be saying that "homorphic" applies to a set while "analytic" necessarily applies to all of C. That's not any notation I have ever seen. A function is "entire" if it is analytic on all of C.
     
  19. Jul 18, 2006 #18
    Atleast I have not found any refrence to support it. Since a function is holomorphic if it is has a complex derivative it was easy to cut it down to also mean the same for real valued functions. Even more so because of that nice little function that is infinitely differentiable but not analytic. Though as shmoe reprimand, that is not the idea idea I have had.

    I think it does. We have just 'agreed' that i cannot use the term holomorphic for real functions, so the point of making a diffrence between "holomorphic" and "analytic" is not an issuse anymore.

    I hope I am not saying that :)
     
  20. Jul 18, 2006 #19

    shmoe

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    This is also a definition for analytic complex functions. So:

    I would still say this is no explanation for considering them different. Your attempt to apply holomorphic to real functions essentially just changed 'complex' to 'real', yet you already agree to do something different when it comes to the term 'analytic'. This seems like one of the most arbitrarily confusing way of doing things, and serves no purpose whatsoever that I can see.
     
  21. Jul 18, 2006 #20

    Jex

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    I know you might be curious about "holomorphic" and what not but don't you think we are taking this beside the point? :smile: Some of us reading this are not that advanced in mathematics to even understand what some of you are saying. Which is not a bad thing at all, I will get there eventually but right now I am nearing the end of my diff eq class and really the discussion going on here is just confusing me worse on my original question. I didn't know my question that I thought was pretty simple was going to strike a mini debate on terms. So to go back to my original post, what do some of you have to say to the following on analycity:

    "In calculus it is seen that functions such as cosx and ln(x-1) can be represented by power series by expansions in either Maclaurin or Taylor series. It is said that these functions are analytic."

    So basically this is my original question. Is this a strong definition (a reasonable definition for someone who has only taken up to diff eq)? Without getting wrapped up in complex numbers, whether or not it is synonymous with holomorphic, etc.
     
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