### Quick description

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.

### Prerequisites

A basic knowledge of the main theorems of complex analysis.

### Some links

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.

To calculate an integral round a closed path, use theorems rather than direct calculation

**Contour integration tricks**

**Proving that functions are holomorphic**

To prove that a function is holomorphic, just differentiate it

To prove that a function is holomorphic, use Morera's theorem

**Properties of holomorphic functions**

How to show that a holomorphic function with some property must be constant

Some basic facts about holomorphic functions that restrict their behaviour

**Conformal maps**

**Singularities**

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