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At which points (if any) is this function differentiable? At which points is it analytic?

$f(x+iy) = x^2 + iy^2$

I applied the Cauchy Riemann equations and got the result that $y=-x$. So then am I correct to say that the function is only differentiable on the line $y=-x$ and is analytic nowhere as it is not differentiable at every point in any small disc centred on the line?

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State your theorem that you intend to use. Does satisfying C-R at one point imply differentiable at that point? – GEdgar Apr 29 '12 at 18:46
up vote 1 down vote accepted

Hint: Ask yourself: Does it satisfies CR equation at the origin?Is $f$ differentiable at the origin?Is $f$ analytic at the origin? Do you notice it is harmonic function?Your explanation is sufficient: a function is (complex-)analytic at a point only when it is (complex-)differentiable on an open neighborhood of that point, and since your function is (complex-)differentiable only on a line, it is not analytic at any point.only has a complex derivative on the diagonal y = x.

As for the implications of the Cauchy-Riemann equations: they are the additional condition required for a real-differentiable function to be complex-differentiable. As you have figured out, the Cauchy-Riemann equations are a pointwise condition; this problem demonstrates that they may be satisfied at a point without being satisfied in a neighborhood of the point, and in this case the function is complex-differentiable at the point but not analytic there.

If you know about the view of the derivative as a linear transformation, the Cauchy-Riemann equations are exactly the condition that the derivative, a real-linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ should coincide with a complex-linear transformation from $\mathbb{C}$ to $\mathbb{C}$

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