Partial derivative with respect to z & z_bar?

In summary, the author introduces the expressions ∂f/∂z and ∂f/∂z_bar to represent partial derivatives of a complex function of two real variables, f(x,y). The variables z=x+iy and z_bar=x-iy are used as "independent" variables. While there may be confusion about the independence of z and z_bar, they can be treated as independent variables similar to x and y, and z_bar can be fixed while z varies.
  • #1
omyojj
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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?
 
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  • #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.
 
  • #3


Hi there,

The expressions ∂f/∂z and ∂f/∂z_bar are known as the Wirtinger derivatives, and they are used in complex analysis to represent the partial derivatives with respect to z and z_bar. In this context, z and z_bar are considered as independent variables, even though they are not truly independent in the conventional sense.

This is because in complex analysis, z and z_bar are treated as separate variables that can vary independently, even though they are related by the complex conjugate. Therefore, when taking the partial derivative with respect to z, z_bar is treated as a constant and vice versa. This allows us to extend the concept of partial derivatives to complex-valued functions.

So, to answer your question, z and z_bar are considered independent in complex analysis, and the Wirtinger derivatives are used to represent the partial derivatives with respect to these variables. I hope this helps clarify things for you.
 

1. What is a partial derivative with respect to z?

A partial derivative with respect to z measures the rate of change of a multivariable function with respect to the variable z, while holding all other variables constant. It is denoted by ∂f/∂z.

2. How is a partial derivative with respect to z calculated?

To calculate a partial derivative with respect to z, treat all other variables as constants and differentiate the function with respect to z using the usual rules of differentiation.

3. What does a partial derivative with respect to z_bar indicate?

A partial derivative with respect to z_bar measures the rate of change of a complex-valued function with respect to the variable z_bar, while holding all other variables constant. It is denoted by ∂f/∂z_bar.

4. Can a partial derivative with respect to z and z_bar be calculated simultaneously?

No, a partial derivative with respect to z and z_bar cannot be calculated simultaneously as they are independent variables. Each partial derivative must be calculated separately.

5. What is the relationship between partial derivatives with respect to z and z_bar?

The relationship between partial derivatives with respect to z and z_bar is governed by the Cauchy-Riemann equations. For a function to be complex-differentiable, its partial derivatives with respect to z and z_bar must satisfy these equations.

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