*Analytic* is a term that describes a complex function that is "differentiable" at and in the neighborhood (region of non-zero area) surrounding some point. A function that is analytic over the entire complex plane is called entire.

But this is an empty definition unless we clarify what it means for a complex function to be differentiable. Differentiability of a complex function f(z = x+iy) is a very strict criterion. f(z) is defined to be differentiable at a point z_{0} iff the limit as Δz goes to 0 of (f(z_{0}+Δz) - f(z_{0}))/Δz exists and is independent of *how* Δz goes to 0. For example, the limit must be the same for both the case when Δz = Δx and the case when Δz = iΔy. More generally the limit must be the same for all 360 degrees worth of directions of Δz in the complex plane.

The mathematical necessary conditions for a function f(z) to be differentiable at a point are called the Cauchy-Riemann equations, and are very easy to prove. It can be proven that given any analytic function, its first derivative is analytic as well, implying that *every analytic function has derivatives of all orders in the region in which it's analytic*. This statement is not true for differentiable functions of real variables. For instance f(x) = x|x| is differentiable at x = 0, but its derivative is not.

Analytic functions have wonderful mathematical properties, and I am unaware of interest in or work on nonanalytic function theory.