Results 1 to 4 of 4

Math Help - Derivatives and the Cauchy-Riemann equations

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    5

    Derivatives and the Cauchy-Riemann equations

    I need to decide whether the following statement is true or not:
    let f be a complex function, f(z)=f(x+iy)=u(x,y)+iv(x,y).if u and v satisfy the Cauchy-Riemann equations in a neighborhood of z_0 then f is differentiable at z_0.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    False.

    This is Mine 75th Post!!!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by Liav Teichner View Post
    I need to decide whether the following statement is true or not:
    let f be a complex function, f(z)=f(x+iy)=u(x,y)+iv(x,y).if u and v satisfy the Cauchy-Riemann equations in a neighborhood of z_0 then f is differentiable at z_0.
    no. consider f(z) = u(x,y) + iv(x,y), where u(x,y) = \sqrt{|xy|}, v(x,y) = 0 and consider the derivative at z_0 = 0

    i got this example from a text book.

    can you come up with a nicer example, TPH?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2007
    Posts
    5

    I don't think the example is correct:

    "if u and v satisfy the Cauchy-Riemann equations in a neighborhood of z_0" - It means there's an open circle (disk) around z0 where the Cauchy-Rieamann equations are satisfied. In your example there is no such circle (neighborhood). Every circle around (0,0) in your example contain positive real numbers where it is easy to show the Cauchy-Riemann equations aren't satisfied.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. When exactly do we use the Cauchy-Riemann equations?
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 18th 2011, 02:47 AM
  2. The Cauchy Riemann equations and harmonic functions
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: March 15th 2011, 06:45 PM
  3. Cauchy-Riemann equations
    Posted in the Calculus Forum
    Replies: 5
    Last Post: December 21st 2009, 08:14 PM
  4. Cauchy Riemann equations
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 6th 2009, 03:54 PM
  5. Cauchy Riemann Equations
    Posted in the Calculus Forum
    Replies: 6
    Last Post: January 3rd 2009, 08:53 AM

Search Tags


/mathhelpforum @mathhelpforum