Is f(z) := x^2 + iy^2 Holomorphic Anywhere?

[blahblah8724]

If z = x + iy then the function f(z) := x^2 + iy^2, has total derivative,

\begin{pmatrix} 2x & 0 \\ 0 & 2y \end{pmatrix}

so surely by the Cauchy–Riemann equations this is complex differentiable at x = y, but is this function holomorphic anywhere?

Thanks!

 

[HallsofIvy]

No, it is not analytic anywhere. In order to be analytic at a point, the Cauchy-Riemann equations must be satisfied in some neighborhood of the point and that does not happen here.

By the way, the phrase "holomorphic anywhere" does not make sense. "Holomorphic" means "analytic for all complex numbers". You mean to ask if it was analytic anywhere.

 

[Citan Uzuki]

By the way, the phrase "holomorphic anywhere" does not make sense. "Holomorphic" means "analytic for all complex numbers". You mean to ask if it was analytic anywhere.

? According to the Wikipedia article, holomorphic at a point means complex-differentiable in some neighborhood of the point. So asking whether a function is holomorphic anywhere does make sense.