Often a specific technical term is coined in mathematics because a concept is so often repeated. The usual modern presentation method of Definition- Theorem - Proof often starts with the discovery method of Object - Property - Pattern, just finding and naming things is the first step.
In another thread, I came across the concept of 'holomorphic'. An answer to How to express in closed form? led to my asking the question Complex conjugate of z without knowing z=x+iy.
I had never heard the term 'holomorphic' before, but wikipedia helped. Looking at complex analysis texts online, I couldn't find anything in their indexes. Lots of papers use it, and there are lots of questions here on math.SE, so it must be part of everyday (higher) math. The definition includes 'analytic' which (I think ) means you can take an infinite sequence of derivatives (which means you can create a Taylor series for it)). But then I don't see what's special about that (I'm sure that's obvious but I just don't know; so you can compute values quickly?) and that means something even -more- for complex functions.
So the question here is...why is 'holomorphic' such a big deal? And is it in fact? If a complex function is holomorphic you can do ..what with it? if it is not holomorphic, is there anything else you can do with it? What specific areas of mathematics deal with it? is it simply basic complex analysis or part of some particular branch? (I'm actually not even sure that it is particular to complex analysis)
(My background is not analysis at all (cs, combinatorics, logic) so I know basic real analysis but not complex at all.)