what is real analysis and complex analysis
In: Mathematics
Mathematical analysis is generally a formal way of studying stuff. In particular, looking at the numbers themselves, sequences, series, functions, limits, continuity and so on.
Basically it looks at how things work.
Real analysis limits itself to just real numbers. Complex analysis does the same stuff but lets you use complex numbers as well.
Analysis is about comparing things (majorizing and minorizing), functions, calculus, &c.
The real numbers and the complex numbers can be used in a very similar manner, but their *topologies* are quite different: the reals form a line and the complexes a plane. In other words a neighborhood of a point looks, generally speaking, like an interval on the real line and like a disk in the complex plane.
This has deep consequences.
For example you can directly compare two reals but there is no order relation between complex numbers which makes analysis more involved (e. g. working with their moduli – the “lengths”, if you want).
But the main difference is in calculus: while the derivative of a real function need not even be continuous, the existence of the derivative of a function in the complex sense makes that function automatically *analytic*, i. e. representable by a power series, its Taylor series, and having derivatives of all orders. (On the other hand there are real functions with a perfectly defined full Taylor series at a point, but which are different from it.)
TL; DR: both are “analysis”, but the objects studied and the tools of the trade are very different.
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