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Mapping Analytic Functions

  1. Jan 18, 2008 #1
    1. The problem statement, all variables and given/known data
    This is an example in Advanced Engineering Mathematics by Erwin Kreyszig p.675 which I dont understand. If you map [tex]w=z^2[/tex] using Cartesian Co-ordinates, w is defined as
    [tex]w=u(x,y)+iv(x,y)[/tex], therefore, [tex]u=Re(z^2)=x^2-y^2[/tex] and [tex]v=Im(z^2)=2xy[/tex]. The function is graphed using u and v as the axes, and a line x=c is graphed as a parabola as is the like y=k.

    What I want to understand is, that is this so because the surface we were graphing these lines on (which was the xy plane) has been warped in such a manner as to define a new plane uv so that the projection of the lines x=c and y=c, on this uv plane turns out to be a parabola? Is that so?
     
  2. jcsd
  3. Jan 18, 2008 #2

    Dick

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    w=z^2 is a nonlinear function. It's going to change lines in the z plane into curves. I'm not sure why this would surprise you. So, yes, it is so.
     
  4. Jan 19, 2008 #3
    Thank you. Is my explanation right? The second paragraph about the space being warped?
     
  5. Jan 19, 2008 #4

    Dick

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    Well, yes, because the mapping of z->w is nonlinear, if that's what you mean by 'space being warped'. If you've shown they are parabolas then I think you are done.
     
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