What are similarities between real and complex analysis? I am self studying complex analysis from my lecture notes and textbook Ponnusamy and Silvermann I had done a course on complex analysis but the instructor was not interested in teaching so I had to self study and was on my own.
But the  the bigger picture and comparison with real analysis was not clear.
So, I tried googling and found some nice articles on difference between real and complex analysis here:
Differences between real and complex analysis?
http://data.conferenceworld.in/ESM/P246-252.pdf
and a lecture of Ben Brubaker (Univ. Minnesota) in complex analysis.(It was the first lecture)
But I cannot find anything about similarities between real and complex analysis?
Can you please shed some light on what are similarities between real and complex analysis in terms of results/ theorems which are similar?
I shall be really thankful
 A: I will answer this from a pedagogical point of view. The first couple of real analysis courses are very focused on rigorously proving pretty simple things involving sequences, subsequences, sets, limits, series, continuity, differentiable functions, Mean value theorem, and Riemann integrals. Almost every proof involves playing around with epsilons and the definitions.
If there's anything one should take away from complex analysis, it's the connection between analytic functions and power series. Almost everything, including Cauchy's integral formula, Cauchy's theorem, etc. is connected to this concept. There are nifty, more advanced topics like winding numbers and residue theory.
Complex analysis will also touch on important ideas that connect to differential equations, such as maximum principle (in particular, an analytic function consists of harmonic real and imaginary parts, respectively). Although not my field, complex analysis also covers a number of valuable theorems in algebra involving roots of polynomials, such as the fundamental theorem of algebra.
In my opinion, a rigorous real analysis course is harder. Typically, epsilons and deltas take a back seat in a complex analysis course, where the focus is on new concepts. Sure, limits are frequently involved, but using, for example, the squeeze theorem will not typically involve a rigorous proof.
Complex analysis is a beautiful subject. Good luck.
