Complex Analysis-Difference between Differentiable and Analytic

[gbean]

Homework Statement

Show that f(z) = x^2 + i(y^2) is diff at all points on y=x. Then show that is not analytic anywhere.

Homework Equations

Cauchy Riemann equations: fy = ifx <=> function is differentiable (I'm still unclear about the implications of CR-equations. It says in my book that if f is differentiable at z, fx and fy exist and satisfy the CR-equations, but it also says that a polynomial is analytic <=> CR-equations are satisfied).

The Attempt at a Solution

So I took the partial derivatives:

fy = 2iy
ifx = 2ix

The only way for fy (2iy) = ifx (2ix) is if x=y. This shows that the equation is differentiable because it satisfies CR-equations.

But I'm not sure how to show that this is not analytic. As I understand it, analyticity is a neighborhood property, so just because the function is differentiable at the points where y=x, there doesn't exist a neighborhood around each point where f is differentiable, and so f(z) is not analytic. But is this explanation sufficient to show that a f(z) is not analytic, or do I need to show more?

Also, I thought that satisfaction of CR-equations meant that the function was analytic...But the book says that it is not analytic, despite CR being satisfied on the line y=x.

 

[ystael]

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.

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}.)