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Complex analysis, i.e. the analysis of functions involving one complex-valued variable, definitely deserves a separate part of BookofProofs. One reason for this is that the material dealt with in this part has various applications in mathematics and in particular physics. But probably the most astonishing reason for this is that analysis of functions with complex-valued variables allows getting a deeper understanding of the analysis of functions with real-valued variables. That’s why complex analysis is often referred to as the function theory.
Function theory is a self-closed theory with a kind of mathematical beauty, not least because of it has many vivid geometrical interpretations, in particular of the behavior of points in the complex plane under certain complex-valued functions. The key concept of function theory is that of a holomorphic function. Holomorphy is a property of a function with far-reaching and astonishing consequences. These include:
If a function is holomorphic, then it can be differentiated infinitely many times.
Similarly, a holomorphic function can be integrated infinitely many times (locally) in its domain, and everywhere in its domain, if this domain is path-connected.
If all the infinitely many derivatives of two holomorphic functions are equal at just a single point, then they must be identical everywhere in their domains.
If you integrate a holomorphic function along a closed curve, no matter how “wild” this closed curve might look like – the integral will be always $0$.
A holomorphic function, which is constant in a small neighborhood of a point, is constant everywhere.
If a holomorphic function is bounded everywhere, then it must be constant everywhere (Liouville’s theorem).
A holomorphic function maps open sets into open sets and connected regions into connected regions,
and many more,…
Theoretical minimum (in a nutshell)
In order to start studying function theory, you should be acquainted with: