1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook