<s>[INST] check if a function is holomorphic Let $f:\mathbb{C} → \mathbb{C}$ be defined as
$$f(z)= \begin{cases}\frac{x^3y(y − ix)}{x^6 +y^2}&\text{ if }z = x + iy= 0\\ 0 &\text{ if }z=x+iy=0\end{cases}$$
And holomorph at $z = 0$
I tried to apply the definition of holomorphic application, which is as follows:
$\displaystyle \lim_{(a, b) \to (0,0)} \frac{f (z_0 + h) - f (z_0)}{ h}$
Defining:
$f(x, y) = (xy^2 /x^4 -1, -x^2 y / x^4 -1)$
$f(x + a, y + b) = ((x + a) (y + b)^2 / (x + a)^4 -1, - (x + a)^2. (y + b) / (x + a)^4 -1)$
$f (z_0) = f (x, y) = (xy^2 / x^4 -1, -x^2 y / x^4 -1)$
$ h = (a, b)$
My question is how to solve this limit ...
My question is to define the limit of homologous functions or Verify if it is differentiable from a real point of view and use the cauchy rieman equations.
Thank you!
 [/INST] Set $f=u + iv$, so $u=\frac{xy^2}{x^4-1}$ and $v=\frac{-x^2y}{x^4-1}$.
calculating the partial derivatives we have:
$$\frac{\partial u}{\partial x} = \frac{y^2(x^4-1)-xy^2(4x^3)}{(x^4-1)^2}$$
And
$$\frac{\partial v}{\partial y}= \frac{-2xy(x^4-1)-(-x^2y)(4x^3)}{(x^4-1)^2}$$
as the first equation is already different, it follows that f is not holomofa at 0.
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