What are the conditions for a complex function to be analytic in a domain? Provide a definition of analytic function.

A function $f(z)$ is said to be analytic (or holomorphic) at a point $z_0$ if its derivative $f'(z)$ exists not only at $z_0$ but also at every point in some neighborhood of $z_0$. A function is analytic in an open set if it is analytic at every point in that set.

Let $f(z) = u(x,y) + iv(x,y)$ be a complex function, where $z = x+iy$.
The necessary and sufficient conditions for $f(z)$ to be analytic in a domain $D$ are:

1. The