Results 1 to 3 of 3

Math Help - Analytic vs differentiable

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    107
    Thanks
    2

    Analytic vs differentiable

    Hi,
    I need some help with the question attached.

    It's my understanding that an analytic function is differentiable at every point in the domain.


    What does "nowhere analytic" mean. Is the premise behind this question that the function is differentiable on the coordinate (does coordinate mean real?) axes and not differentiable on the imaginary axes?

    OR, is it that, there is some point where the function is differentiable, but it is not an entire function?


    i found the C-R Equations.

    du/dx = 3x^2 + 3y^3 -3 , dv/dy = 3y^2 - 2 + 3x^2

    du/dy = 6xy , dv/dx = 6xy

    du/dx = dv/dy

    and du/dy = - dv/dx where x = 0 or y = 0

    and thus C-R equations are not satisfied on an open disk, but only on the line x=0 or y=0

    I think this is wrong.
    Any help appreciated, especially if you point out the thought process required here.

    Thanks.
    Attached Thumbnails Attached Thumbnails Analytic vs differentiable-healpz.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,569
    Thanks
    1410

    Re: Analytic vs differentiable

    Quote Originally Posted by 99.95 View Post
    Hi,
    I need some help with the question attached.

    It's my understanding that an analytic function is differentiable at every point in the domain.


    What does "nowhere analytic" mean. Is the premise behind this question that the function is differentiable on the coordinate (does coordinate mean real?) axes and not differentiable on the imaginary axes?
    The point is that a function that is differentiable at every point on an open set (the "open disc" that you refer to) is analytic on that set. And, no "coordinate" does not mean "real". You used the phrase "coordinates axes" which is plural! Both real and imaginary axes are "coordinate axes".

    OR, is it that, there is some point where the function is differentiable, but it is not an entire function?


    i found the C-R Equations.

    du/dx = 3x^2 + 3y^3 -3 , dv/dy = 3y^2 - 2 + 3x^2

    du/dy = 6xy , dv/dx = 6xy

    du/dx = dv/dy

    and du/dy = - dv/dx where x = 0 or y = 0

    and thus C-R equations are not satisfied on an open disk, but only on the line x=0 or y=0

    I think this is wrong.
    Any help appreciated, especially if you point out the thought process required here.

    Thanks.
    Last edited by HallsofIvy; August 25th 2013 at 03:45 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2013
    From
    United States
    Posts
    18
    Thanks
    2

    Re: Analytic vs differentiable

    I think you have your answer in front of you already. Your thought process is right. The C-R eqns. are satisfied only if the point is on either of the coordinate axes which means that the function is differentiable only on those points which are on one of the coordinate axes. However the function is not differentiable in any open disc in the complex plane. (Try to pick one!) So it not analytic at any point in the complex plane.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. differentiable but not analytic function
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 9th 2012, 08:45 AM
  2. Replies: 0
    Last Post: March 5th 2012, 04:50 PM
  3. Replies: 9
    Last Post: December 17th 2010, 08:13 AM
  4. Replies: 0
    Last Post: October 3rd 2010, 07:03 AM
  5. Sequence of differentiable functions, non-differentiable limit
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 3rd 2009, 05:13 AM

Search Tags


/mathhelpforum @mathhelpforum