Z (conjugate) not analytic?

  • Thread starter 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.
 

Answers and Replies

  • #2
Dick
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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.
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
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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
Dick
Science Advisor
Homework Helper
26,260
619
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.
 

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