Complex analysis is the study of complex-differentiable functions. In a first course on complex analysis, these functions are normally defined on domains, that is, connected open subsets of . The definition of a complex-differentiable function looks just the same as the definition of a differentiable function from to , but this appearance is misleading: complex-differentiable functions are so unlike real-differentiable functions that they are given a different name. They are known as holomorphic functions. Holomorphic functions can also be defined on Riemann surfaces and complex manifolds. At a more advanced level, there are many connections between complex analysis and geometry. More generally, complex analysis is a tool that is used throughout mathematics.
A basic knowledge of the main theorems of complex analysis.
At the moment there are very few complex analysis articles on the Tricki, so here is a list of all of them. Later, it will almost certainly be necessary to add another level to the hierarchy. In fact, a start has been made on that process, by grouping together two articles into a general "contour integration" heading.
Contour integration tricks
Proving that functions are holomorphic
Properties of holomorphic functions