Complex analysis front page

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Complex analysis front page

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[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 $\C$. The definition of a complex-differentiable function looks just the same as the definition of a differentiable function from $\R$ to $\R$, 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 [[w:Riemann surfaces]] and [[w: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.

[note attention] This is a hastily-put-together page in order to give a parent to the few articles on complex-analysis tricks we have so far.[/note]

===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 a contour integral, use theorems rather than direct calculation|To calculate an integral round a closed path, use theorems rather than direct calculation]]

'''Contour integration tricks'''
*[[Shift the contour of integration]]
*[[If you are calculating an integral, complex substitutions are not forbidden]]

'''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'''
*[[How to find a conformal map from one domain to another]]

'''Singularities'''
*[[When thinking about singularities, you often don't need to expand in a power series]]
*[[How to calculate residues]]

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