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Derivative of Complex Variables

  1. Oct 19, 2011 #1
    Hi,

    What is the following derivative:

    [tex]\frac{\partial}{\partial x}|b-ax|^2[/tex]?

    Now I know that [tex]|b-ax|^2=(b-ax)(b^*-a^*x^*)[/tex], so how to do the differentiation with respect to [tex]x^*[/tex]?

    Thanks in advance

    PS.: All variables and constants are complex.
     
  2. jcsd
  3. Oct 20, 2011 #2

    Mark44

    Staff: Mentor

    Is this a homework question?
     
  4. Oct 20, 2011 #3

    lurflurf

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    Homework Helper

    It is not clear what that derivative means.
    A guess would be that it is a Wirtinger derivative in which case we have
    [tex]\frac{\partial x^*}{\partial x}=0[/tex]
     
  5. Oct 20, 2011 #4
    No it is not.

    So lurflurf, are you saying that the derivative will be:

    [tex]-a(b-ax)^*[/tex]

    I thought it will be like:

    [tex]-(b-ax)a^*[/tex]

    but I couldn't prove it.
     
  6. Oct 20, 2011 #5

    lurflurf

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    Homework Helper

    Again it is not clear what that derivative means, but a good guess would be -a(b-ax)*.
    Then we have
    d|b-a x|^2=-a(b-a x)* dx+-a*(b-a x) dx*

    notice that |b-a x|^2 is definitely not complex differentiable as it depends upon x and x* rather than upon x alone
     
  7. Oct 20, 2011 #6
    No, I just need to the derivative [tex]\frac{\partial}{\partial x}[/tex], where the derivative is partial with respect to x. I see some books writing x as a+jb, and then compute the derivatives with respect to a and b. I was just wondering if there is another way to do this.

    Thanks
     
  8. Oct 20, 2011 #7

    lurflurf

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    Homework Helper

    There are different ways because a complex variable can change in more ways than a real variable. We can work with different coordinates or none. So a function of a complex variable can be described in diferent ways with two variables
    |x| and arg(x)
    Re(x) and Im(x)
    x and x*
    and so on

    since you are interested in the x partial x and x* are natural, but you could use any set and the chain rule to find the x partial.
    [tex]\frac{\partial}{\partial x}=\frac{\partial u}{\partial x}\frac{\partial}{\partial u}+\frac{\partial v}{\partial x}\frac{\partial}{\partial v}[/tex]
    However you want to choose u and v
     
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