Analytic function

A function $f(z)$ is analytic at a point $z_0$ if it is complex differentiable of all $z$ in the neighbourhood of $z_0$. An alternate definition (which the above leads from and is more rigorous) is that a function is analytic if it is locally given by a convergent Taylor series. A function is entire if it is analytic for all $z\in \mathbb{C}$.

By theorem, if $u$, $v$ and their partial derivatives are continuous on a domain $D$ (open connected set) and satisfy the Cauchy-Riemann equations, then $f=u+iv$ is analytic in $D$.

If a function and its partials are continuously differentiable everywhere, then they are analytic everywhere. Also by theorem, if $f$ is analytic in $D$, then $u$ and $v$ are harmonic in $D$. If $f$ is analytic, then $u$, $v$ have continuous derivatives of all orders.

As a more geometric interpretation, if $f$ is analytic in $D$, then $\nabla u\cdot \nabla v=0$ at all points $(x,y)\in D$, which follows from the Cauchy-Riemann equations (expand it! — it will simplify nicely). Since the gradient is perpendicular to level curves, the functions $u$ and $v$ intersect orthogonally.

Also geometrically, if $f$ is analytic within the region (and it makes up a vector field in ${\mathbb{R}}^2$ where $x=u$, $y=v$), then the vector field has zero curl and zero divergence.

$$\nabla \cdot \vec{F}=0$$ $$\nabla \times \vec{F}=0$$