partial derivative with respect to z & z_bar??

omyojj

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?

HallsofIvy

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.