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Homework Help: Complex question

  1. Mar 27, 2010 #1
    1. The problem statement, all variables and given/known data
    show that the function f(z)=zRe(z) is only differentiable at the origin.

    im completely lost with is, its probably very easy.. but dont know how to start.
    f(x+iy)=x+iy*Re(z)=x(x+iy)??? idk
  2. jcsd
  3. Mar 27, 2010 #2
    I believe you need to use the Cauchy-Riemann equations, u_x = v_y, v_x = -u_y, where
    u is the real part of the function and v is the imaginary part, and u_x is the derivative of u with respect to x etc.

    As you stated, the function can be written as x(x+iy) = x^2 + ixy, so u = x^2, v = xy. You will find the functions only satisfy the C-R equations at the origin.
  4. Mar 27, 2010 #3


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    Gold Member

    So far so good, now what are the real and imaginary parts, [itex]u(x,y)[/itex] and [itex]v(x,y)[/itex], of this expression? What conditions must the partial derivatives of [itex]u(x,y)[/itex] and [itex]v(x,y)[/itex] satisfy for f(x+iy) to be differentiable?

    EDIT: NT123 beat me to it. :smile:
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