# Thread: Complex differentiable and holomorphic functions

1. ## Complex differentiable and holomorphic functions

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

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

f is differentiable only at $\displaystyle z_{0}=0$

For g, solving for x and y in Cauchy-Riemann I end up with $\displaystyle 6y^{2}=6y^{2}$ and $\displaystyle xy=-\frac{1}{3}$ so $\displaystyle z_{0}$ is in the form $\displaystyle z_{0}=x-\frac{1}{3x}i$. Now, $\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=\frac{2}{3x^{2}}$ are continuous $\displaystyle \forall x \in \mathbb{R}$\{$\displaystyle 0$} and since $\displaystyle x=\frac{-1}{3y}$ we must have $\displaystyle x,y \in \mathbb{R}$\{$\displaystyle 0$}. So, g should be differentiable $\displaystyle \forall z_{0} \in \mathbb{C}$\{$\displaystyle 0$} satisfying $\displaystyle x=\frac{-1}{3y}$.
But I have read that polynomial with coefficients in $\displaystyle \mathbb{C}$ are differentiable in $\displaystyle \mathbb{C}$. Hence, I'm not sure about my answer for g.

h is differentiable only at $\displaystyle z_{0}=0$

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

f and h are nowhere holomorphic.

if my answer for the first part is right, then g is holomorphic in $\displaystyle (-\infty , 0)$ and $\displaystyle (0, \infty)$. But now that I think of it it doesn't really make much sense.

2. Notice that g is not really a polynomial in $\displaystyle \mathbb{C}$. As for the other question, a function f is holomorphic at z if it's (complex) differentiable in an open ball around z.

3. Originally Posted by Jose27
Notice that g is not really a polynomial in $\displaystyle \mathbb{C}$.
I'm not sure I see why g is not a polynomial in $\displaystyle \mathbb{C}$.

As for the other question, a function f is holomorphic at z if it's (complex) differentiable in an open ball around z.
I know the definition of holomorphic but I'm not sure if I understand how to apply it.

Do my answers to both questions seem right to you? What about the differentiability of g?

4. Notice that if then is not a lower bound for .
So .
In this way construct a decreasing sequence of distinct terms from converging to .

Consider this set of nonnegative numbers: .

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