Is Real Analysis Necessary Before Complex Analysis?

[Howers]

Just wondering, when starting on introductory analysis is it logical to do real analysis before complex variables? My guess is complex analysis uses things from real analysis. I'm doing very basic analysis in calc 2, and not sure if its enough to get by complex.

 

[Johan de Vries]

You can study complex analysis in parallel. There are some very powerful tools that can be derived from complex analysis. If you like to do integrals, then you will certainly enjoy contour integration methods that can be used to compute ordinary real integrals of functions that don't have an antiderivative expressible in terms of elementary functions.

Just yesterday I explained to a physics student who didn't have any knowledge about complex analysis how to compute the integral of the function cos(ax)/cosh(x) from zero to infinity using contour integration. Of course, I had to explain the method in a heuristic way, but at least he understands how to use the method.

As I explained on another thread, letting students see some powerful results may motivate them to study the theory in detail...

 

[mathwonk]

there are two levels of calculus, first you do calculus of smooth functions, and then later in rel analysis they also study much less well behaved functions,a nd their integrals.by definition complex analysis is about functions which are even better than smooth, they are analytic. thus the basic complex analysis course is comoarable to the first course in calculus where you are studying really well behaved functions, but functions with complex values instead of real.

hence from this point of view complex analysis is more elementary tham real analysis. after learning basic complex analkysis, one can proceed further and incorporate ideas from more avanced real analysis. for this point of view see rudins real and complex analysis. or check out hormanders several complex variables, chapter one.

but you don't need much real analysis to do beginning or even intermediate complex analysis, just path integration and power series.

 

[SiddharthM]

Any analysis in Calc2 would have to be very elementary. Although many Complex Analysis courses at the undergraduate level aren't rigorous (for example: many that use Brown&Churchill) enough to REALLY be considered analysis. In fact some such courses are more similar to calculus classes in that they're more concerned with applications than pure theory, with techniques than proofs.

Looking at Complex Variables and Applications by Brown & Churchill I'm beginning to realize that to complete many of these excersizes you only REALLY need calculus 2 and multivariable calculus (not even, this is just to ensure you've gotten used to the idea that the plane can be a domain).

This all being said I believe it's a good idea to take a stiff Advanced Calculus course (functions of one real variable, introduction to analysis, are other names for it) that emphasizes rigorous mathematics BEFORE a complex analysis course. The theory of complex functions is a powerful and elegant but mathematical maturity is needed to appreciate this.

Also Real Analysis is a misleading term - it can mean the first analysis course that one takes which is often called Advanced Calculus. It can also mean a more advanced graduate level theory which is also called Measure theory and is about generalizing the idea of the 'length'/"size" of a set of real numbers. This course cannot be taken without first taking Advanced Calculus.