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How to find the equation of a line in complex analysis?

  1. Jan 21, 2009 #1
    *This is not homework, though a class was the origin of my curiosity.

    In real analysis we could find the equation of a line that passes through two points by finding the slope and then plugging in one set of points to calculate the value of b. ie

    [tex]y = mx + b[/tex]

    m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

    In complex analysis, we know that the equation for a line is [tex]Re[((m+i)z+b)]=0[/tex]. Sitting down to derive m, I find the following:

    [tex]m = \frac{Im[z_1] - Im[z_2]}{Re[z_1]-Re[z_2]}[/tex]

    But if I try to plug in the points (say [tex]z_{1}[/tex] and [tex]z_{2}[/tex]), it doesn't give me a value for b that makes sense. what is the correct way to find the equation of a line?
     
    Last edited: Jan 21, 2009
  2. jcsd
  3. Jan 21, 2009 #2
    The expression you gave is not an equation. What is the complete expression?
    The complex plane is simply an imaginary axis and a real axis at right angles to each other. If z = (x, y) = x + iy is an arbitrary complex variable, it pretty much replaces y in our equations with Im[z] and x with Re[z]. so if you have a graph with points (x, y) that satisfy the equation y = mx + b, you will get the same graph in the complex plane with the equation Im[z] = m*Re[z] + b. This is not very geometric, however, and your equation for the slope of a line passing through the points z1 and z2 is more descriptive:
    I'm not sure what equation you're plugging it into but geometrically, a line can be defined as the set of points equidistant from two distinct points in the plane, say z1 and z2. This gives us the equation |z - z1| = |z - z2| for the set of points z on the line.
    If you go ahead and translate this into the point-slope form, you get your m as above and the equation b = [[Re(z2)]22 + [Im(z2)]2 - ([Re(z1)]22 + [Im(z1)]2)]/2(Re(z2) - Re(z1)) = [|z2| - |z1|]/2(Re(z2) - Re(z1)) .
     
  4. Jan 21, 2009 #3
    Yes, that makes sense now. I think I was just crunching wrong. I amended my previous equation. [tex]Re((m+i)z + b))=0[/tex]

    If I let [tex]z_1=1+0i[/tex] and [tex]z_2=-1-i[/tex]

    I calculate:

    [tex]m=\frac{0-(-1)}{1-(-1)}=\frac{1}{2}[/tex]

    [tex]Re((\frac{1}{2}+i)*(1+0i)+b))=0[/tex]
    [tex]Re(\frac{1}{2}+i+b)=0[/tex]
    [tex]b=-\frac{1}{2}[/tex]

    And the equation of the line is: [tex]Re((\frac{1}{2}+i)z - \frac{1}{2})=0[/tex]

    This works. Thank you!
     
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