Ask HN: Can you suggest advanced books on integral calculation?
I enjoy calculating hard integrals and would like to master it. I just started with my master in physics, so I would love to see some "next-level" techniques.
30 comments
I could suggest the following
Inside Interesting Integrals, by Nahin. This contains integrals with often arithmetic inspiration. Imagine things like, lots of special values of the zeta function or partial sums of harmonic numbers and that sort of thing.
Irresistible Integrals, by Boros and Moll. This is inspired by their work to prove all of the integrals in the enormous book of Gradshteyn and Ryzhik, whose references are often old and lacking (like Erdelyi's prior table of integrals).
You might also like Solved Problems for the Gamma and Beta Functions, Legendre Polynomials, and Bessel functions by Farrel and Ross. This is a bit closer to real analysis and veers towards more practical integral magic.
Finally, I'll note that all of these largely omit complex residue calculus (using complex analysis to solve real integrals). I don't know of a good book specifically aimed at this, unfortunately.
Inside Interesting Integrals, by Nahin. This contains integrals with often arithmetic inspiration. Imagine things like, lots of special values of the zeta function or partial sums of harmonic numbers and that sort of thing.
Irresistible Integrals, by Boros and Moll. This is inspired by their work to prove all of the integrals in the enormous book of Gradshteyn and Ryzhik, whose references are often old and lacking (like Erdelyi's prior table of integrals).
You might also like Solved Problems for the Gamma and Beta Functions, Legendre Polynomials, and Bessel functions by Farrel and Ross. This is a bit closer to real analysis and veers towards more practical integral magic.
Finally, I'll note that all of these largely omit complex residue calculus (using complex analysis to solve real integrals). I don't know of a good book specifically aimed at this, unfortunately.
Also:
(Almost) Impossible Integrals, Sums, and Series - Cornel Ioan Vălean
(Almost) Impossible Integrals, Sums, and Series - Cornel Ioan Vălean
Besides the suggestions below, it is instructive to go to the math stackexchange and look at the highest voted Q&As tagged 'integrals' or 'definite-integrals'. Legends like Cleo lurk there demonstrating mind-boggling feats of integration. The corresponding mathoverflow tags are also interesting, but likely to involve more advanced concepts.
As practical advice (not addressing your enjoyment), learn how to use a CAS effectively. That will have a ton more bang for the buck in studying physics than being able to occasionally impress yourself with a Feynman trick or obscure application of the King property.
what CAS do you recommend? Are the open source alternatives at the same level as licensed ones? I know Mathematica from Wolfram is well used in the territory.
And how does one learn how to use those?
I personally use Maxima and Sage. They're not as polished as Mathematica by a long shot, but they're free and very useful when it you get enough practice with them.
Along a similar line, the only book that I've found that has goes through a careful derivation of the divergence theorem using Lebesgue integration is, "A Concise Introduction to the Theory of Integration," Second Edition by Daniel W. Stroock. I prefer this one to his later book, "Essentials of Integration Theory for Analysis."
Does anyone else have a good reference or book on the topic?
Does anyone else have a good reference or book on the topic?
Also when you talk about the divergence theorem using Lebesuge integration, you're tending into the territory of Geometric Measure Theory which is very deep and im not sure why anyone would do this unless they were a working mathematician and had a practical need for it, like say they were studying geometric analysis (which only a small group of people in the world do).
This is an amazing book and I'm so suprised to see someone else knows about it. However
, who cares about the Lebesgue integral? The only thing its good for is integrating pathological functions like the rationals, the indicator of the Cantor set, and fractals. Riemann integration is just fine and I'm not sure what all the fuss is about the Lebesgue Integral. Sure expectation of a random variable is a Lebesgue integral, but most of the time you have a density anyways and you can use the Riemann Integral.
As mentioned in a sibling comment, Lebesgue integration can be helpful with probability theory because we can wrap some information into the measure rather than the function. Though, to be sure, this can often be done in a similar manner using the Riemann-Stieltjes integral.
To me, part of the value of Lebesgue integration is in understanding the limitations of Riemann integrals and when they break. Some of this is covered in Stroock's book in chapter 5.1. Alternatively, when in working in function spaces, we may need to integrate in a more general way than Lebesgue integration, so things like Bochner integrals, which require similar theory. This can arise in the theory related to things like PDE constrained optimization, which most of the time is targeted toward physics related models.
All that said, bluntly, I prefer to work with Riemann integrals when at all possible. However, the same question then applies. Do you or someone else have a reference for a rigorous derivation of the divergence theorem or integration by parts in multiple dimensions using Riemann integration? It's not particularly hard in one dimension, but higher dimensions is tricky and it's hard to get the details of integrating on the surface correct. Stroock's book is the only reference that I know of and he does it with Lebesgue integration.
To me, part of the value of Lebesgue integration is in understanding the limitations of Riemann integrals and when they break. Some of this is covered in Stroock's book in chapter 5.1. Alternatively, when in working in function spaces, we may need to integrate in a more general way than Lebesgue integration, so things like Bochner integrals, which require similar theory. This can arise in the theory related to things like PDE constrained optimization, which most of the time is targeted toward physics related models.
All that said, bluntly, I prefer to work with Riemann integrals when at all possible. However, the same question then applies. Do you or someone else have a reference for a rigorous derivation of the divergence theorem or integration by parts in multiple dimensions using Riemann integration? It's not particularly hard in one dimension, but higher dimensions is tricky and it's hard to get the details of integrating on the surface correct. Stroock's book is the only reference that I know of and he does it with Lebesgue integration.
Well the mathematicians care for those pathological cases. Probably the most important of said pathological cases is Brownian motion / Weiner processes / SDEs. Brownian motion is differential with zero probability, yet it has many modeling applications. It is also fractal-like (the self-similarity property).
The more practical advantage measure theory provides for probability is you can simultaneously handle continuous and discrete distributions. Most of the time what works for one works for the other but you can get some weird mistakes (Shannon’s differential entropy has a few issues as a measure of information not found in the discrete case because he got lazy and just replaced the sum with an integral).
A good chunk of the time I come across measure theoretic probability papers I feel like they’re making the paper a lot more complicated and messier than it needs to be, but it does serve a purpose.
The more practical advantage measure theory provides for probability is you can simultaneously handle continuous and discrete distributions. Most of the time what works for one works for the other but you can get some weird mistakes (Shannon’s differential entropy has a few issues as a measure of information not found in the discrete case because he got lazy and just replaced the sum with an integral).
A good chunk of the time I come across measure theoretic probability papers I feel like they’re making the paper a lot more complicated and messier than it needs to be, but it does serve a purpose.
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"A Course of Higher Mathematics" - Smirnov
"Differential and Integral Calculus" - Piskunov
"Problems in Mathematical Analysis" - Demidovich (exercises/problems)
"Differential and Integral Calculus" - Piskunov
"Problems in Mathematical Analysis" - Demidovich (exercises/problems)
Advanced Calculus by Woods.
https://zackyzz.github.io/feynman.html
https://zackyzz.github.io/feynman.html
"Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals" - Boros and Moll
"Inside Interesting Integrals" - Nahin
"Inside Interesting Integrals" - Nahin
Learn complex analysis contour integrals! There super fun for doing physics integrals. I'm not sure if these are too basic for what your asking about or not, but thought I'd mention them. Junjiro Noguchi's Introduction to Complex Analysis is a book recommended from herehttps://math.stackexchange.com/questions/438468/what-is-the-...
It's fantastic that you enjoy tackling challenging integrals, especially as you embark on your master's in physics! To master advanced techniques, consider exploring resources like "Advanced Calculus" by Patrick M. Fitzpatrick, or delving into specific areas like contour integration for complex analysis.
Since you are already doing this, can you suggest some good books you have be using for beginners. ?
I really liked the calculus book by Swokowski. The books starts with a revision of the requirements to start learning derivatives, has sections dedicated to applications of that and then goes on to integrals, analytical geometry and differential equations. It will keep you entertained for months.
The AP Calculus material on Khan Academy is excellent. One thing they could add however is more material on the integration of hyperbolic functions (which isn't part of the core AP material).
I've been looking at Foundation Maths by Croft and Davison as a refresher for everything math, which includes calc
Get Gradshteyn and Ryzhik and follow the references ...
https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik
https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik
It might be a bit disappointing, but there aren’t much in the way of next level techniques.
Sure there might be one off tricks to solve specific problems, but for integration these tend not to be useful in general.
Sure there might be one off tricks to solve specific problems, but for integration these tend not to be useful in general.
> Sure there might be one off tricks to solve specific problems
There is a widely applicable generic algorithm for indefinite integration (https://en.wikipedia.org/wiki/Risch_algorithm).
> but for integration these tend not to be useful in general.
That is true. For many problems, simpler methods work.
Also, according to that page “Currently, there is no known full implementation of the Risch algorithm”.
There is a widely applicable generic algorithm for indefinite integration (https://en.wikipedia.org/wiki/Risch_algorithm).
> but for integration these tend not to be useful in general.
That is true. For many problems, simpler methods work.
Also, according to that page “Currently, there is no known full implementation of the Risch algorithm”.
The MIT integration bee has some good problems. So do the Putnam tests.
I remember study guides for the first actuarial exam give tricks to calculate repeated integrals fast.
There's also those CRC manuals filled with solutions to integrals that show the steps.
There's also those CRC manuals filled with solutions to integrals that show the steps.
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