# Thread: Holomorphic functions

1. ## Holomorphic functions

(a) At what points $z \in \mathbb{C}$ are the functions $f(z)=|z|^4$ and $g(z)=g(x+iy)=6xy^{2}+i(4x+2y^{3})$ and $h(z)=z \overline{z}^{2}$ 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 $z_{0}=0$

For g, solving for x and y in Cauchy-Riemann I end up with $6y^{2}=6y^{2}$ and $xy=-\frac{1}{3}$ so $z_{0}$ is in the form $z_{0}=x-\frac{1}{3x}i$. Now, $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=\frac{2}{3x^{2}}$ are continuous $\forall x \in \mathbb{R}$\{ $0$} and since $x=\frac{-1}{3y}$ we must have $x,y \in \mathbb{R}$\{ $0$}. So, g should be differentiable $\forall z_{0} \in \mathbb{C}$\{ $0$} satisfying $x=\frac{-1}{3y}$.
But I have read that polynomial with coefficients in $\mathbb{C}$ are differentiable in $\mathbb{C}$. 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?

2. 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 $f(z)=a_0 z^n+a_1 z^{n-1}+\cdots+a_n$ 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 $f(x,y)=g(x,y)+ih(x,y)$ is complex analytic iff h(x,y) is the complex congugate of g(x,y).