#algebra 

**Holomorphic (or analytic, or regular) function** in some domain is a [[Комплексные числа|complex]]-valued function that is [[Cauchy-Riemann equations|differentiable]] in every point of that domain

## Consequences
If a function is differentiable in some domain, then its real and imaginary parts are [[Harmonic function|harmonic functions]]:
$$\LARGE \Delta u=0, \Delta v=0$$Using the Cauchy-Riemann equations and the fact that our functions are smooth
![[Pasted image 20251024232517.png]]


The real and imaginary parts of a holomorphic function can't have [[Максимум и минимум функции|maximums or minimums]] in the complex plane - just the saddle
**Critical point**: $$\LARGE z_0: \frac{df}{dz}\Big|_{z=z_0}=0$$
Critical point of a complex function is the critical point of its real and imaginary parts
![[Pasted image 20251024232527.png]]