Results 1 to 3 of 3

Math Help - Complex differentiation

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    64
    Thanks
    4

    Complex differentiation

    Hey, I have this problem from a tute i was given,
    Its all in the picture,

    For part a i just showed the derivatives exist and that they were continuous, then used laplace to show that they are harmonic du/dx - du/dy = 0,
    so that work, then to find v i just found the harmonic conjugate and i got -3y^2x+x^3 + c = v
    so u+iv = y^3 - 3x^2y +i(3y^2x+x^3 + c )
    Is that the complete solution to the question? I feel like I am missing somthing, more of a proof or somthing.

    Part b i showed in the picture, i think the first part seems ok, then showing for elsewhere i just let z0 be any point, and tried to find if there were any points besides zero where the limit exists but lim z0/delz is infinity, so is that sufficient to say that that its only differentiable at zero?
    We did somthing different aswell, let it be z= u+iv, then f(z) = u^2 + ivu cause Re(z) = u, then let these u^2 and vu be two new variables and applied C-R using implicite differentiation and found somthing different, that it is differentiable when Re(z) = constant or Im(z) = 0

    Am i on the right track at all?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: Complex differentiation

    Quote Originally Posted by Daniiel View Post
    Hey, I have this problem from a tute i was given,
    Its all in the picture,

    For part a i just showed the derivatives exist and that they were continuous, then used laplace to show that they are harmonic du/dx - du/dy = 0,
    so that work, then to find v i just found the harmonic conjugate and i got -3y^2x+x^3 + c = v
    so u+iv = y^3 - 3x^2y +i(3y^2x+x^3 + c ) Note the missing minus sign.
    Is that the complete solution to the question? I feel like I am missing something, more of a proof or something.
    If z = x+iy and f(z) = u+iv, then I think they want you to express f(z) as a function of z (rather than x and y). With the change of sign that I indicated, you have f(z) = ix^3 - 3x^2y - 3ixy^2 + y^3. Can you see how to write that as a function of z? [Think: Binomial theorem.]

    Quote Originally Posted by Daniiel View Post
    Part b i showed in the picture, i think the first part seems ok, then showing for elsewhere i just let z0 be any point, and tried to find if there were any points besides zero where the limit exists but lim z0/delz is infinity, so is that sufficient to say that that its only differentiable at zero?
    We did somthing different as well, let it be z= u+iv, then f(z) = u^2 + ivu cause Re(z) = u, then let these u^2 and vu be two new variables and applied C-R using implicite differentiation and found somthing different, that it is differentiable when Re(z) = constant or Im(z) = 0

    Am i on the right track at all?
    The first part of b) is ok. For the second part, you are right to use the CR equations, but you are getting the notation a bit mixed. If z = x+iy then f(z) = x^2 + ixy. So you should take u = x^2 and v = xy. Then apply the CR equations.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2008
    Posts
    64
    Thanks
    4

    Re: Complex differentiation

    oh sweet thanks very much, so in terms of z for a, f(z)= iz^3 +ic right?
    So for part b, because both C-R equations don't hold it is not differentiable elsewhere? only if x=y=0
    Last edited by Daniiel; September 30th 2011 at 05:36 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex Differentiation
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 17th 2010, 04:58 PM
  2. Complex Variables differentiation
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: October 11th 2009, 12:46 PM
  3. Differentiation of Complex Numbers
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 20th 2009, 05:55 AM
  4. differentiation with complex variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 20th 2008, 10:06 AM
  5. differentiation and complex numbers
    Posted in the Calculus Forum
    Replies: 10
    Last Post: August 15th 2007, 08:36 PM

Search Tags


/mathhelpforum @mathhelpforum