1. In mathematics: the abstract study of characteristic properties possessed by different classes of mathematical functions.
2. In Systems Engineering
a. The prescriptive process of decomposing functions into smaller functions.
b. The descriptive process of identifying functions performed by some existing system or item.
2.b is the original sense of the term in System Engineering, used in the context of developing incremental component-level upgrades (see Hall’s book of the same title).
As used in the modern SE sense, meaning 2.a is actually more synthesis than analysis.