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Complex function

  1. Oct 24, 2009 #1
    1. The problem statement, all variables and given/known data

    How can i determine whether a complex function has any local maximum or minimum?

    2. Relevant equations

    let's consider the case f(z)=z* (conjugate of z)
    z=x+iy

    3. The attempt at a solution

    f(z)=z*=x-iy

    how do i see if it has local max or min?

    Thank you
     
  2. jcsd
  3. Oct 24, 2009 #2

    Mark44

    Staff: Mentor

    [tex]f(z)~=~z\bar{z}~=~(x + iy)(x - iy)~=~x^2 + y^2[/tex]
    So f(z) is purely real, for any complex number z. It looks to me like it has an absolute minimum but no maximum.
     
  4. Oct 24, 2009 #3
    how do you know it has an absolute minimum?
     
  5. Oct 24, 2009 #4

    Mark44

    Staff: Mentor

    x and y are real numbers. Both x2 and y2 are always nonnegative, so their sum will also always be nonnegative.
     
  6. Oct 24, 2009 #5
    i agree, but why does this have anything to do with max/min?
     
  7. Oct 24, 2009 #6

    Mark44

    Staff: Mentor

    The minimum value of your f(z) is 0 and there is no maximum value. Are you asking about complex functions in general, or about this one?
     
  8. Oct 24, 2009 #7
    im asking in general.
    if we look at your example, f(z)=zz*
    then why is there a minimum?
     
  9. Oct 24, 2009 #8

    Mark44

    Staff: Mentor

    If you're asking in general, then you would be looking at the derivative and seeing where it's zero, and testing the critical points.

    You're overthinking your example (it's not mine). You have f(z) = zz* = x2 + y2, where z = a + bi. For this particular function, it's very easy - almost trivial - to discover that the minimum value is 0 (for z = 0 + 0i), and that the function is unbounded. For any real numbers x and y, x2 [itex]\geq[/itex] 0 and y2 [itex]\geq[/itex] 0, which means that x2 + y2 [itex]\geq[/itex] 0.

    If you think about it graphically, the complex plane is the domain and the image of the function is a paraboloid that opens upward and whose vertex is at (0, 0).
     
  10. Oct 24, 2009 #9
    oh, so the graph would be like y=x^2 in real coordinate system
    ?

    how did you know that f=x^2+y^2 looks like that in the complex plane?
     
  11. Oct 24, 2009 #10

    Mark44

    Staff: Mentor

    Not in the complex plane. The complex plane is the domain.
    I know what it looks like because I know what the graph of z = x2 + y2 looks like in R3. The only thing different is that the domain in one of these is the complex plane and in the other it's the real x-y plane.
     
  12. Oct 24, 2009 #11
    ok, so it's in 3d. I get it now, so we have: z=x^2+y^2 not f(z)=x^2+y^2
    right?
     
  13. Oct 24, 2009 #12

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In the original post, you asked "How can i determine whether a complex function has any local maximum or minimum?".

    In general, the question doesn't make any sense because the complex numbers aren't ordered. Which is larger, 3 + 4i or 4 + 3i?

    You have given a particular function that happens to have real values, which is why your particular problem makes sense.
     
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