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Graphing complex functions(the image)

  1. Nov 13, 2007 #1
    I'm really unsure how to go about graphing a complex function. Like, f(z) = z^2, where z = x+iy.

    This ISN'T a homework problem, but I'm studying for an exam and that's an example in a book I'm reading and it says "the image of this function" and goes on explaining some things relevant to the drawing, but there doesn't seem to be a systematic way to go about doing this.

    Thanks.
     
  2. jcsd
  3. Nov 15, 2007 #2
    Find the real and imaginary images of the mapping.

    Since [tex] z= x+iy[/tex] then [tex]z^2 = x^2-y^2+2ixy[/tex].

    Thus we have [tex]\Re(z^2) = x^2-y^2 \text{ and } \Im(z^2) = 2xy[/tex]

    Now think of this as a mapping from [tex]\mathbb{R}^2 \rightarrow \mathbb{R}^2[/tex] under the function [tex]f(x,y) = (x^2-y^2, 2xy)[/tex]
     
  4. Nov 15, 2007 #3

    HallsofIvy

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    Which means, of course, that you would need a four-dimensional graph!

    What is often done is to take u(x,y)+ iv(x,y)= f(z)= x+ iy. Draw some lines in an xy-plane and show what those are mapped into in the uv-plane.
    For example, with f(z)= f(x+iy)= (x2-y2+ i(2xy), the
    horizontal line y= 0 is mapped into u= x2, v= 0 which is just the vertical line v= 0. The horizontal line y= 1 is mapped into u= x2-1, v= 2x so u= v2/4, a parabola.
     
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