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Differentiating complex numbers

Hi
I'm having some difficulty with this simple question:
'Determine for what values of z (if any) the function f(z) = |z| is differentiable.'

My attempt:
[br]z=x+iy[br]∣z∣=x2+y2∈R[br]whichisdifferentiableatpointswherex2+y2>0 [br]z=x+iy[br]\lvert{z}\rvert=\sqrt{x^2+y^2} \in \mathbb{R}[br]which is differentiable at points where x^2+y^2 >0
Is this right? It seems too simple. Have I misunderstood the question?
Reply 1
Avatar for Taurus
Taurus
12
modulus of z is root X^2 - Y^2 as i^2 = -1

Then I'm not sure, but I'm thinking you would differentiate implicitly by the chain rule if thats possible.
Is this FP2?
Reply 2
Avatar for nuodai
nuodai
17
Original post by Taurus
modulus of z is root X^2 - Y^2 as i^2 = -1

Then I'm not sure, but I'm thinking you would differentiate implicitly by the chain rule if thats possible.
Is this FP2?


That's not right, the modulus of x+iyx+iy is x2+y2\sqrt{x^2+y^2}. It's the distance from the origin (in the Argand diagram) of x+iyx+iy, which corresponds to the point (x,y)(x,y).

By the looks of it this is first (maybe second) year undergraduate; differentiability definitely isn't in A-level!

Original post by Muffin.
Hi
I'm having some difficulty with this simple question:
'Determine for what values of z (if any) the function f(z) = |z| is differentiable.'

My attempt:
[br]z=x+iy[br]∣z∣=x2+y2∈R[br]whichisdifferentiableatpointswherex2+y2>0 [br]z=x+iy[br]\lvert{z}\rvert=\sqrt{x^2+y^2} \in \mathbb{R}[br]which is differentiable at points where x^2+y^2 >0
Is this right? It seems too simple. Have I misunderstood the question?


You just seem to have stated the answer without proving it. Plug it into the definition of differentiability and take some limits to justify your claim.
(edited 12 years ago)
Reply 3
Avatar for thiccc
thiccc
17
nooooo please dont say maths gets this hard! :frown:
Reply 4
Avatar for Taurus
Taurus
12
Original post by nuodai
That's not right, the modulus of x+iyx+iy is x2+y2\sqrt{x^2+y^2}. It's the distance from the origin (in the Argand diagram) of x+iyx+iy, which corresponds to the point (x,y)(x,y).

oh crap I knew that....
Reply 5
Avatar for Bobifier
Bobifier
13
Have you done the Cauchy-Riemann differential equations?

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