Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

  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


    User Avatar
    Science Advisor

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