#algebra 

## [[Предел|Limit]] and convergence in $\LARGE \mathbb{C}$
$\LARGE a=\displaystyle\lim_{a\rightarrow\infty} z_n$
$\LARGE \forall \varepsilon>0: \exists n_0: \forall n>n_0: |z_n-a|<\varepsilon$ 

## Definition of [[Производная|derivatives]] in $\LARGE \mathbb{C}$
$$\LARGE f'(z_0)=\frac{f(z)-f(z_0)}{z-z_0}\Big|_{z\rightarrow z_0}$$
$$\LARGE \Delta f=\Delta u+i\Delta v=\frac{\partial u}{\partial x}\Delta x+\frac{\partial u}{\partial y}\Delta y+i(\frac{\partial v}{\partial x}\Delta x+\frac{\partial v}{\partial y}\Delta y)$$
We can group terms 1 4 and 2 3 if the following apply:
$$\LARGE \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$
$$\LARGE \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$
These are the [[Cauchy-Riemann equations]] (or conditions)