Z (conjugate) not analytic?


by Applejacks
Tags: analytic, conjugate
Applejacks
Applejacks is offline
#1
Oct24-11, 10:12 PM
P: 33
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.
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Dick
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#2
Oct24-11, 10:24 PM
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Quote Quote by Applejacks View Post
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.
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
Applejacks is offline
#3
Oct24-11, 10:41 PM
P: 33
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
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#4
Oct24-11, 10:51 PM
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P: 25,175

Z (conjugate) not analytic?


Quote Quote by Applejacks View Post
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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