Thread: Holomorphic functions

(a) At what points are the functions and and differentiable? At what points are f and g and h holomorphic?

Using Cauchy-Riemann equations (+ showing continuity of partial derivatives) I have found:

f and h are differentiable only at

For g, solving for x and y in Cauchy-Riemann I end up with and so is in the form . Now, are continuous \{ } and since we must have \{ }. So, g should be differentiable \{ } satisfying .
But I have read that polynomial with coefficients in are differentiable in . Hence, I'm not sure about my answer for g.


As to where the functions are holomorphic, I'm not quite sure I understand the concept very well. This is what I have found:

f and h are nowhere holomorphic.

I am not sure how to approach this for the function g.

Can someone please check these answers for me?

Holomorphic means differentiable in a neighborhood of a point. The functions (a) and (c) are not differentiable in a "region" about a point.

As for (b) don't confuse a "complex polynomial" with a "real polynomial" with complex coefficients. Now, the complex polynomial is entire but your function is not of that form but rather just two real functions with an "i" coefficient in front of the second one.

And the function is complex analytic iff h(x,y) is the complex congugate of g(x,y).