Why is the function f(z) = ¯z not differentiable for any z ∈ C?

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In summary, being "Z (conjugate) not analytic" means that a function is not analytic in the complex plane when its variables are replaced with their complex conjugates. This is different from being non-analytic, which can occur in other ways. The significance of this condition is that it helps identify certain types of non-analytic behavior and understand the concept of analytic continuation. A function can be "Z (conjugate) not analytic" at some points and analytic at others, and the Cauchy-Riemann equations can be used to determine if a function meets this condition.
  • #1
Applejacks
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Homework Statement



Show that f(z) = ¯z is not differentiable for any z ∈ C.

Homework Equations


The Attempt at a Solution



Is it because the Cauchy-Reimann Equations don't hold?

Z (conjugate) = x-iy
u(x,y)=x
v(x,y=-iy

du/dx=1≠dv/dy=-1
du/dy=0≠-dv/dx=0
Edit: Is there another approach? Because the CR Equations is something we learned later on.
 
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  • #2
Applejacks said:

Homework Statement



Show that f(z) = ¯z is not differentiable for any z ∈ C.

Homework Equations


The Attempt at a Solution



Is it because the Cauchy-Reimann Equations don't hold?

Z (conjugate) = x-iy
u(x,y)=x
v(x,y=-iy

du/dx=1≠dv/dy=-1
du/dy=0≠-dv/dx=0
Edit: Is there another approach? Because the CR Equations is something we learned later on.

Sure. Use the definition of f'(z)=lim h->0 (f(z+h)-f(z))/h. Show the limit is different if you pick h to be real from the limit if you pick h to be imaginary. That's really what the content of the CR equations is.
 
Last edited:
  • #3
I think I get it now.

(f(z+h)-f(z))/h

(conjugate((z+h)-z))/h = h(conjugate)/h

If h=Δx, the ratio equals 1
If h=Δiy, the ratio equals -1.

Since the two approaches don't agree for any z, z(conj) is not analytic anywhere. Correct?
 
  • #4
Applejacks said:
I think I get it now.

(f(z+h)-f(z))/h

(conjugate((z+h)-z))/h = h(conjugate)/h

If h=Δx, the ratio equals 1
If h=Δiy, the ratio equals -1.

Since the two approaches don't agree for any z, z(conj) is not analytic anywhere. Correct?

Yep, that's it. That's how you derive CR.
 

1. What does it mean for a function to be "Z (conjugate) not analytic?"

Being "Z (conjugate) not analytic" means that a function is not analytic in the complex plane when its variables are replaced with their complex conjugates. In other words, the function is not differentiable at any point where the complex variables are replaced with their conjugates.

2. How is a function being "Z (conjugate) not analytic" different from being non-analytic?

Being "Z (conjugate) not analytic" is a specific condition that only applies to functions with complex variables. A function can be non-analytic in other ways, such as being discontinuous or having a singularity, without being "Z (conjugate) not analytic."

3. What is the significance of a function being "Z (conjugate) not analytic?"

This condition is important in complex analysis because it allows for the identification of certain types of non-analytic behavior in functions. It also helps to define and understand the concept of analytic continuation, which is the process of extending the domain of a function by using its analytic properties.

4. Can a function be "Z (conjugate) not analytic" at some points and analytic at others?

Yes, this is possible. A function can exhibit "Z (conjugate) not analytic" behavior at some points in the complex plane while still being analytic at others. For example, a function may be non-analytic at the origin but analytic everywhere else.

5. How can one determine if a function is "Z (conjugate) not analytic?"

To determine if a function is "Z (conjugate) not analytic," one can use the Cauchy-Riemann equations. These equations state that a function is analytic at a point if and only if its partial derivatives with respect to the real and imaginary parts of the variable exist and are continuous at that point. If these conditions are not met, then the function is "Z (conjugate) not analytic."

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