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Complex operator in polar coords

  1. Sep 1, 2011 #1
    1. The problem statement, all variables and given/known data
    If z=x + iy, what is d/dz in polar coordinates?

    3. The attempt at a solution

    I know that expanded,

    d/dz = 1/2 (d/dx) - i (d/dy)

    Where to go from there?
  2. jcsd
  3. Sep 2, 2011 #2


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    Science Advisor

    Use the chain rule:

    You have
    [tex]\frac{df}{dz}= \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}i[/tex]
    [tex]= \left(\frac{\partial f}{\partial r}\frac{\partial r}{\partial x}+ \frac{\partial f}{\partial \theta}\right)+ \left(\frac{\partial f}{\partial r}\frac{\partial r}{\partial y}+ \frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial y}\right)i[/tex]

    with, of course, [itex]r= \sqrt{x^2+ y^2}[/itex] and [itex]\theta= arctan(y/x)[/itex].
  4. Sep 4, 2011 #3
    Why is "f" necessary there?

    Also, should there be a "partial theta partial x" in the first parentheses?
    Last edited: Sep 4, 2011
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