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Simple complex function question

  1. Aug 15, 2007 #1
    Find f(z)= u(x,y) + iv(x,y), given U [tex] = x^2 - 2xy - y^2 \\ [/tex] and check for analyticity.
    We have to find v(x,y) as follows:
    [tex] u_x = v_y [/tex] and [tex] u_y = -v_x [/tex] Cauchy-Riemann equations
    [tex] u_x = 2x - 2y [/tex] and [tex] u_y = -(2x+2y) \\[/tex].
    Therefore[tex]v_y = 2x - 2y [/tex]......(i)
    and [tex] v_x = 2x + 2y \\[/tex] .......(ii), integrating (i) with respect to y and then differentiating it with respect to x , we v=[tex] 2xy - y^2 +[/tex]h(x) and [tex] v_x = 2y + \frac{dh}{dx} \\[/tex] on comparision with (ii) [tex] \frac{dh}{dx} = 2x [/tex] therefore h(x)= [tex] x^2+c \\[/tex] Therefore [tex] v = 2xy - y^2 +x^2+c\\[/tex]. Question. How can h(x) be a constant of integration, I thought the constant of integration could only be a pure number?
    Last edited: Aug 15, 2007
  2. jcsd
  3. Aug 15, 2007 #2


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    If you are integrating and differentiating wrt y, then sure, h(x) can be considered a constant of integration. It's derivative wrt y is 0. Of course, it can't wrt x. It's all relative.
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