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I'm struggling with the exercises in Stein and Shakarchi's Complex Analysis, and believe what I'm missing is advanced techniques (as opposed to concepts) in real analysis.

The difficulty I consistently have is not with the new material introduced regarding complex analysis, but with techniques assumed (not discussed or presented) of real analysis. Examples:

  1. Replace $\frac{1}{z-a}$ with an infinite geometric series
  2. Replace $\sin \theta$ with $2\theta/\pi$ using Jordan's inequality
  3. Replace $1 - e^{iz}$ with the power series of $e$ which tends to $-iz$ as $z \to 0$ (assumed on p.45 of text, explained on math.SE, though I can't find the link)
  4. and many other similar cases

All these examples have in common:

  • The proof or problem applies a non-trivial technique to go from one step to the next, usually to replace an intractable expression with a tractable one
  • The technique is not discussed in the text
  • The technique applies to real analysis (not complex)
  • The technique is a technique*, as opposed to understanding a concept

How, then, can I build my real analysis techniques so that I can follow Stein and Sharakachi?

I know real analysis well enough to solve all the problems on a MIT OCW real analysis final (although sometimes with difficulty), but not well enough to solve the problems on a Rudin based course final. This might suggest to learn real analysis again, using Rudin. However, looking at the Rudin text and course, I don't see them teaching these advanced techniques but rather more advanced concepts, such as point-set topology and metric spaces beyond the line.

What course should I use to learn the techniques needed to approach Stein and Shakarachi, then?

Or should I simply work through Stein and Shakarachi, and use it as a motivator to look up (and learn) new techniques?

*By technique I mean a means of simplifying an expression that does not follow directly from the underlying definitions and theorems, but instead is a creative application from elsewhere (such as those discussed here.) I distinguish technique from concept, which means understanding the objects, definitions, and theorems. Both of them, of course, are crucial.

I'm not the first person to raise this issue. But the recommendation given there (to first learn complex variable calculus before learning the proofs) seems to miss the point, as the issue, as shown by those examples, is not with complex variables but with techniques from real analysis.

SRobertJames
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  • Look at the exceptional book "Visual Complex Analysis" by T. Needham. – Jean Marie Oct 24 '22 at 21:13
  • @JeanMarie That may be a great book, but it teaches the geometric intuition behind complex analysis. My goal is to learn the advanced techniques of real analysis, enough to follow the proofs and do the problems of Stein and Sharakachi. – SRobertJames Oct 24 '22 at 21:23
  • "advanced techniques of real analysis" but you write also "advanced techniques of complex analysis" in the title of your question... – Jean Marie Oct 24 '22 at 21:25
  • "for complex analysis": the techniques needed for application to complex analysis, but seem to come from real. Regardless, does Needham teach technique, like the examples I described? Or simply build geometric intuition? – SRobertJames Oct 24 '22 at 21:27
  • He does both. Besides, there are books on real analysis that are very enlighting. I like in particular "Fourier analysis" by T. Körner which is very easy to read. Have a look at Körner's delicious homepage here – Jean Marie Oct 24 '22 at 21:35
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    "By technique I mean a means of simplifying an expression that does not follow directly from the underlying definitions and theorems" yet it looks like $e^z$ is defined in chapter 1, page 14 of your book in terms of the power series $\sum_{n=0}^\infty \frac{z^n}{n!}$ . So your example (3) seems to be a case of applying the definition not applying a technique per se. Perhaps the issue is manipulating absolutely convergent power series which is something one typically gets in real analysis. – user8675309 Oct 24 '22 at 22:16
  • @user8675309 Yes, $e^z$'s series is certainly known. But it didn't occur to me to say "Well, $1 - e^z$ is close enough to $-iz$ for small $z$ that I can replace it and integrate it in the limit". Nor was it obvious how they made that switch, even though I'm very familiar with $e$'s power series. – SRobertJames Oct 24 '22 at 23:49
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    In practice you may not even know if a problem can be solved when you're trying to solve it. I would encourage you to keep struggling. – CyclotomicField Nov 08 '22 at 22:18

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