(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Where is f(z) differentiable? Analytic?

[tex]f(z) = x^{2} + i y^{2}[/tex]

2. Relevant equations

Cauchy-Riemann Equations

3. The attempt at a solution

I calculated the partial derivatives,

[tex]u_{x} = 2x[/tex]

[tex]v_{y} = 2y[/tex]

[tex]u_{y} = 0[/tex]

[tex]v_{x} = 0[/tex]

Then said that for the CR equations to hold,

[tex]u_{x}=v_{y}[/tex] therefore [tex]y=x[/tex]

and

[tex]u_{y}=-v_{x}[/tex] therefore [tex]0=0[/tex]

Then becuase the partial derivatives are continuous for all [tex]x,y[/tex], [tex]f(z)[/tex] is differentiable along [tex]y=x[/tex]

[tex]f(z)[/tex] is nowhere analytic because an arbitrarily small open disk centered at any point on the line [tex]y=x[/tex] will always contain points which are not differentiable.

Is that sufficient to show differentiability? Or am I misapplying the cauchy-riemann conditions?

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# Homework Help: CR equations and differentiability

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