Results 1 to 5 of 5
Like Tree2Thanks
  • 1 Post By HallsofIvy
  • 1 Post By Plato

Thread: Proving a Function is Analytic

  1. #1
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    666
    Thanks
    2
    Awards
    1

    Proving a Function is Analytic

    I would like to prove that this function is analytic. It is really hard for me to see.

    We suppose that f is analytic and zero free in a domain D, the function I wish to prove is analytic is:

    \int_{z_0}^z \frac{f'(\zeta )}{f(\zeta )} ~d\zeta

    My professor tells me there is a standard argument where I take a z_1 \in D and a small disk D_r(z_1) \subset D then take a fixed path from z_0 to z_1 and then a line segment from z_1 to z, but I am not sure where to go from there.

    Any help is greatly appreciated!
    Last edited by Aryth; Apr 14th 2016 at 12:03 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    18,947
    Thanks
    2737

    Re: Proving a Function is Analytic

    You understand that this integral, over any path, is just ln\left(\frac{f(z)}{f(z_0)}\right), don't you?
    Thanks from Aryth
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    666
    Thanks
    2
    Awards
    1

    Re: Proving a Function is Analytic

    Quote Originally Posted by HallsofIvy View Post
    You understand that this integral, over any path, is just ln\left(\frac{f(z)}{f(z_0)}\right), don't you?
    I was not aware that the integral was path independent. I was told that this integral is path independent if and only if there is a branch of the logarithm of f. If we have two paths \gamma_1 and \gamma_2 from z_0 to z with \gamma = \gamma_1 \circ \gamma_2^{-1} then,

    \int_{\gamma_1} \frac{f'(\zeta )}{f(\zeta )} ~d\zeta - \int_{\gamma_2} \frac{f'(\zeta )}{f(\zeta )} ~d\zeta = \int_{\gamma} \frac{f'(\zeta )}{f(\zeta)} ~d\zeta = 2\pi i \  n(f\circ \gamma , 0) \in 2\pi i  \mathbb{Z}

    which is zero if and only if there is a branch of the logarithm of f.

    Am I missing something?
    Last edited by Aryth; Apr 14th 2016 at 01:35 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,008
    Thanks
    2547
    Awards
    1

    Re: Proving a Function is Analytic

    Quote Originally Posted by Aryth View Post
    I was not aware that the integral was path independent. I was told that this integral is path independent if and only if there is a branch of the logarithm of f.
    You are given "f is analytic and zero free in a domain D".
    I assume that by domain we know that D is path-wise connected.
    There is a subdomain $D_0\subset D~\&~z_0\in D_0$. What do you know about the analyticy of $\log(f(z))~?$
    Thanks from Aryth
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    666
    Thanks
    2
    Awards
    1

    Re: Proving a Function is Analytic

    Quote Originally Posted by Plato View Post
    You are given "f is analytic and zero free in a domain D".
    I assume that by domain we know that D is path-wise connected.
    There is a subdomain $D_0\subset D~\&~z_0\in D_0$. What do you know about the analyticy of $\log(f(z))~?$
    (I apologize for the late reply, I have been away from the internet for a moment).

    Yes, that is what I meant by a domain.

    I was told that a branch of a logarithm exists if and only if the domain is simply connected (i.e. no holes). So, would that mean that this integral is the logarithm locally? Say, in an open disc?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving that an analytic function on a region is constant.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 18th 2011, 05:42 PM
  2. analytic function
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Feb 24th 2011, 01:36 AM
  3. analytic function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Apr 18th 2010, 04:28 PM
  4. analytic function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 18th 2008, 09:57 PM
  5. Proving analytic trig
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Mar 11th 2008, 09:20 PM

Search Tags


/mathhelpforum @mathhelpforum