(a) At what pointsare 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 withand
so
is in the form
. Now,
are continuous
\{
} and since
we must have
\{
\{
But I have read that polynomial with coefficients inare differentiable in
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 polynomialis 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 functionis complex analytic iff h(x,y) is the complex congugate of g(x,y).
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