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I am trying to teach myself real analysis, but I was wondering if this subject is just a collection of proofs of calculus theorems. Aside from set theory, I haven’t learned anything new. By that, I mean I haven’t come across any new theorems that I didn’t already know from calculus. Also, in calculus, there are many difficult problems, including challenging integrals and limits. However, the problems I have faced so far in real analysis were hard because of my proof-writing skills, not because they were difficult problems like integrals.

My second question is: If real analysis does not deal with hard integrals and if calculus books like Thomas' book don't have hard problems, then where does the insanely hard integrals that I see online came from ? something like nonelementary integrals or nonelementary function like $\operatorname{Li}(x)$. Where I can study them if not in real analysis or calculus ?

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    $\begingroup$ Real analysis allows calculus to be made rigorous. But it covers a lot more than that (for me, most notably the intersections of real analysis with measure theory). Insanely hard integrals are solved through great creativity but rarely any genuinely subtle analysis techniques that resemble, say, how you prove standard real analysis problems $\endgroup$
    – FShrike
    Commented May 17, 2023 at 20:52
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    $\begingroup$ If all you want is strangely hard integrals, the book Inside Interesting Integrals is a place to go. Otherwise, the idea of real analysis is to understand the foundations of calculus and study some of those edge cases. For example, can you always exchange a derivative and infinite sum? Can you always pass limits through integrals? $\endgroup$ Commented May 17, 2023 at 22:00
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    $\begingroup$ "hard integrals" sometimes come from other fields using them. For example, Li(x) is a function from analytic number theory. $\endgroup$
    – Ted
    Commented May 17, 2023 at 22:53
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    $\begingroup$ Edits to this question and to an answer brought it to the current queue. I saw this question/answers in May, but I don't see any comments by me and I also remember thinking (then) that @fedja's answer is good (apparently I neglected to upvote it -- fixed now). Indeed, your question is somewhat analogous to someone who has only learned grade school arithmetic wondering what kind of harder arithmetic problems higher math deals with. However, since I'm here, I may as well point out things like Real Analysis Exchange (continued) $\endgroup$ Commented Aug 2, 2023 at 16:22
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    $\begingroup$ and some of my MSE answers such as 1 and 2 and 3 and 4 and 5 and 6 and 7 and 8 and 9, and there are many others (also in mathoverflow, Math. Educators, etc.) $\endgroup$ Commented Aug 2, 2023 at 16:23

3 Answers 3

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The beginning of Real Analysis certainly has significant overlap with the curriculum of most Calculus courses, though the emphasis is different. Whereas standard Calculus courses often emphasize the computation of things like limits, derivatives, and integrals, Real Analysis attempts to get students to think more rigorously about many of the concepts taught in Calculus.

"Aside from set theory, I haven't learned anything new" - the point is not to learn the results from Calculus all over again - you already learned the fundamental theorem of calculus, mean value theorem, intermediate value theorem, etc. when you took Calculus the first time. The point is to learn the proof techniques which are commonly used in Analysis. You should be fully comfortable being able to produce an $\epsilon -\delta$ proof all on your own to show things like convergence and continuity. You should feel confident in your understanding of how a metric endows a space with a topology and how this topology affects the properties of functions defined on that space. You should know how to work with different kinds of norms (e.g. supremum and operator norms) to show things like uniform convergence or boundedness. A standard first course in Real Analysis is not Calculus on steroids - it is not there to teach you how to compute more complicated integrals than the ones you see in Calculus - sorry but you won't find the logarithmic integral function $li(x)$ in most introductory Real Analysis books. It is there, instead, to give you the foundations you need to be successful in the study of mathematical objects defined in terms of limits.

"However, the problems I have faced so far in real analysis were hard because of my proof writing skills and not because they were difficult problems like integrals" - arguably, the exercises in Real Analysis have nothing to do with the integrals and everything to do with the proof-writing skills. You're trying to run before you can walk. You want to study special functions like $li(x)$, but the very definition of that function won't make any sense for ($x > 1$) if you haven't studied Complex analysis, and the techniques used in proving complex analysis results will be difficult to understand or employ if you haven't mastered them in real analysis.

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What they teach in Calculus is just the language of real analysis and the most elementary moves. The real problems are more like this:

Let $f$ be a non-negative smooth (say, twice continuously differentiable) function in $\mathbb R^5$ supported on a compact set $K$. Put $U(x)=\int_{\mathbb R^5}\frac{f(x+y)}{|y|^3}\,dy$ and $V=|\nabla U|$ (the absolute value of the gradient of $U$). Is it true that $\max_{\mathbb R^5}V\le C\max_KV$ with some absolute constant $C$?

If you have completed the full Calculus sequence, you should have no trouble understanding the question. Now try to solve it :-)

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  • $\begingroup$ I hardly can understand the problem :-( so that is the difference between them thank you very much $\endgroup$
    – pie
    Commented May 17, 2023 at 23:08
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    $\begingroup$ @Ahmed I gave you a real research question. If you want to look at exercises in real analysis somewhat above the level you encountered so far, I recommend amazon.com/… $\endgroup$
    – fedja
    Commented May 17, 2023 at 23:31
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    $\begingroup$ @Ahmed If you mean problem solving books rather than thick reference ones, unfortunately no :-( $\endgroup$
    – fedja
    Commented May 18, 2023 at 1:28
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    $\begingroup$ @Ahmed Those are abundant. Just google "Special functions book" and you'll get everything from introductory texts to encyclopedias. The real question then is what fits your level and learning objectives. $\endgroup$
    – fedja
    Commented May 29, 2023 at 13:43
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    $\begingroup$ @FShrike Yep, your edit is correct. Thank you! $\endgroup$
    – fedja
    Commented Aug 2, 2023 at 13:23
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I double majored in Physics and Mathematics at University. There are practical uses of principles of real analysis. For example, suppose the wave function of a particle is constant over a finite, continuous region of space. Frequently, if a system has a finite uncertainty in its position, it has a finite uncertainty in its momentum. That is not the case in this scenario and I could only wrap my head around why by using principles from Real Analysis. In solving the problems, one is tempted to swap the order of derivatives, sums, and integration where it actually doesn't work because the necessary condition of Uniform Convergence isn't satisfied.

No doubt, you've heard of Taylor and Fourier series. The Stone-Weierstrass Theorem broadens the classes of functions you can use for such series breakdowns.

More generally, real analysis teaches you some useful approximation techniques allowing you to give a range of values for an integral you can't integrate analytically. For example, consider a SIRS model for a pandemic. It's 4 non-linear coupled differential equations. $\frac{dk}{dt}=c_1+c_2e^{-kq}+c_3k$. This can't be solved in closed form, but suppose you replace $(1-kq)$ for $e^{-kq}$. Now its solvable and implies a value for the arrival of herd immunity. How inaccurate is this result from the actual answer given that you don't know the actual answer? Numerical methods are available, but frequently the stability and accuracy of such methods derives from principles of real analysis.

Chaos Theory is essentially applied real analysis from start to finish.

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  • $\begingroup$ (uniform convergence isn't a necessary condition) $\endgroup$
    – FShrike
    Commented Aug 2, 2023 at 16:45

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