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Partial derivative with respect to z & z_bar?

  1. Nov 6, 2007 #1
    partial derivative with respect to z & z_bar??

    Hi, all..

    While I`m reading the Ahlfors` complex analysis..I`ve found a tricky expressions about partial derivatives..

    On the theory of analytic fns.

    author uses the expressions ∂f/∂z , ∂f/∂z_bar (z_bar - complex conjugate)

    with f=f(x,y)(f is a complex fn of two real variables..)

    by introducing z=x+iy, z_bar=x-iy as new "independent" variables..

    By the way, can z and z_bar be independent? Moreover, if we write f(z,z_bar) instead,

    the expression ∂f(z,z_bar)/∂z seems to be misleading in a sense that the conventional

    definition of partial derivative tells us that z_bar must be fixed while z varies ( which cannot be)

    Can anybody give me an answer for this?
     
    Last edited: Nov 6, 2007
  2. jcsd
  3. Nov 6, 2007 #2

    HallsofIvy

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    Staff Emeritus
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    z and z_bar are as independent as x and y! If z= x+ iy and z_bar= x- iy, then x= (1/2)(z+ z_bar) and y= (1/2)(z- z_bar)(-i).

    And z_bar certainly can be fixed while z varies. Suppose, for example, (x,y)= (2,3) so that z= 2+ 3i and z_bar= 2- 3i. Then we can vary z while z_bar is fixed by letting z vary along the line z= 1+ 3i.
     
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