Z (conjugate) not analytic?

Applejacks

1. The problem statement, all variables and given/known data

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

2. Relevant equations



3. 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.

Dick

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.

Applejacks

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?

Dick

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