1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Staff Emeritus
    Science Advisor

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?