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Is this valid?

  1. Feb 6, 2012 #1
    Considering f(z) = z where z is analytic. z = x + iy.
    f(z) = u + iv = x + iy.
    Hence u=x and v=y.
    Using Cauchy Reimann eqns.
    ∂u/∂x = ∂v/∂y =1 and
    ∂u/∂y = -∂v/∂x where u=x and v=y hence

    ∂x/∂y = -∂y/∂x

    is this relation true in general?
  2. jcsd
  3. Feb 6, 2012 #2
    I'm assuming you mean z is some function z(w) of another complex number w. So Yes, since f(z) = z = z(w) is just the identity function.
  4. Feb 9, 2012 #3
    I meant to ask whether the relation is true in general and not just complex numbers.

    eg: dy/dx = 1/(dx/dy) for 2 Dimensional plane. Just that way is ∂x/∂y = -∂y/∂x
    true for any 2 Dimensional plane?

    Thankyou. :)
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