#algebra 

For a [[Комплексные числа|complex]]-valued function of $\LARGE f(z=x+iy)=u(x,y)+iv(x,y)$ of a single complex variable $\LARGE z=x+iy$, and $\LARGE u, v$ are real differentiable functions of the the real variables $\LARGE x,y$, then $\LARGE f$ is complex [[Производная|differentiable]] at a complex point if and only if the [[Частная производная|partial derivatives]] of $\LARGE u$ and $\LARGE v$ satisfy the Cauchy-Riemann equations:  

$$\LARGE \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$
$$\LARGE \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$

## Certain applications
Knowing an analytic function's real (or imaginary) part, the other one can be restored:
![[Pasted image 20251024233436.png]]
![[Pasted image 20251024233537.png]] 



![[Pasted image 20251024234001.png]]
![[Pasted image 20251024234006.png]]