- #1

NewtonianAlch

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## Homework Statement

State the Cauchy-Riemann equations and use them to show that the function deﬁned by f(z) = |z|^2 is diﬀerentiable only at z = 0. Find f′(0). Where is f analytic?

## The Attempt at a Solution

f(z) = |z|^2 = (x^2 + y^2)

[itex]\frac{du}{dx}[/itex] = 2x, [itex]\frac{dv}{dy}[/itex] = 0

[itex]\frac{du}{dy}[/itex] = 2y, [itex]\frac{dv}{dx}[/itex] = 0

So x and y must equal 0 for the C.R equations to hold, thus z = 0 + 0 = 0

So this proves they are only differentiable when z = 0.

f'(0) = 2|0| = 0

I'm not sure about the analytic part, f(z) = |z|^2 would be a parabola, so if it's only differentiable when z = 0, i.e. at the origin then it can't be analytic, because an e-neighbourhood at that point will yield points not on the parabolic line, therefore it's not analytic. Is this right or wrong?