|Nov6-07, 06:34 AM||#1|
partial derivative with respect to z & z_bar??
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?
|Nov6-07, 09:04 AM||#2|
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.
|Similar Threads for: partial derivative with respect to z & z_bar??|
|[SOLVED] Partial differential with respect to y||Calculus & Beyond Homework||2|
|Hamilton Jacobi partial derivatives with respect to the constants of||General Physics||2|
|Derivative with respect to something else||Calculus||5|
|derivative of U(X(t),t) with respect to t||Calculus||1|
|derivative with respect to time ??||Introductory Physics Homework||3|