Name: Inside Interesting Integrals
ISBN: 9781493912766

Inside Interesting Integrals

What’s the point of calculating definite integrals since you can’t possibly do them all? What makes doing the specific integrals in this book of value aren’t the specific answers we’ll obtain, but rather the methods we’ll use in obtaining those answers; methods you can use for evaluating the integrals you will encounter in the future. This book is written in a light-hearted manner for students who have completed the first year of college or high school AP calculus, and have just a bit of exposure to the concept of a differential equation. Every result is fully derived. If you are fascinated by definite integrals, then this is a book for you.

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siegelzeroOct 2023

"Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals" - Boros and Moll

"Inside Interesting Integrals" - Nahin

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mixedmathOct 2023

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.

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earthicusFeb 2019

Warning: These books are kind of all over the place as far as technical prerequisites go, but they all satisify being heavier than pop-sci and lighter than a textbook. :)

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Aleksandrov et al - Mathematics: Its content, Methods, ad Meaning. Overview of undergrad curriculum by a number of prominent russian mathematicians.

Levi - The mathematical Mechanic. Uses various physical intuitions and scenarios to solve mathematical problems.

Pask - Magnificent Principia. Tour of newtons principia that includes both newtons geometric language and modern vector calculus version.

Stillwell - Elements of Mathematics. Discusses some important classical problems in the history of mathematics taken from 8 subjects: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, logic. Wow!

Yandell - The Honors Class: Hilbert's problems and their solvers. Lots of mini biographies with some information describing each of Hilbert's famous problems.

Lawvere & Schanuel- Conceptual Mathematics. Some basic notions of category theory, geared towards teaching you Lawvere's clarification and generalization of the incompleteness theorems.

Cormen - Algorithms Unlocked. Pop Science version of some of the textbook.

Nahin - Inside interesting Integrals. Lots of fun integrations!

Nahin - He also wrote some books about euler and imaginary numbers which ive partially read, they seemed good.

Pierce - An introduction to information theory; symbols signals and noise. sounds and looks like a dover textbook but is a non-fake pop science book.

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Boyer - The History of Calculus and its Conceptual Development. A bit old, and spends too much time on formalism in the 18th and 19th century, there might be something better out now.

Weyl - Philosophy of Mathematics and Natural Science.

Reichenbach - The philosophy of Space and time.

Klein - Elementary mathematics from the advanced viewpoint. these were written at the beginning of the 20th century so are slightly outdated, but kline was extremely forward-looking in his point of view.

Courant & Robbins - What is Mathemaics. Another classic.

Hilbert & Cohn-Vossen - Geometry and the Imagination. The gold standard of popular mathematics writing. A beautiful book.

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todd8Dec 2016

For those interested in math and particularly in learning how to attack challenging (definite) integrals, I can recommend this book by Paul Nahin: Inside Interesting Integrals [1]. It's well reviewed on Amazon and I enjoyed the writing.

[1] https://www.amazon.com/gp/product/1493912763/ref=oh_aui_sear...

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rudolfwinestockJan 2023

Entire books have been written about curious integrals.

Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, by George Boros https://www.amazon.com/dp/0521796369/

Inside Interesting Integrals, by Paul J. Nahin https://www.amazon.com/Inside-Interesting-Integrals-Substitu...

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todd8May 2020

About 50 years ago I was fascinated to see a program that could do symbolic integration. Now, programs like Mathematica can do much harder, more complex integrals.

Professor Patrick Winston pointed out that most of the AI programs we were going to look at in his class (chess, natural language, 3D vision in block world, etc.) ended up, like integration, as simple code plus a database of facts.

Back then, there was general optimism about the future of AI. No one anticipated how slow progress would be while the capabilities of the hardware grew a million-fold.

The work referenced in the article is interesting. It appears to be another small step in advancing AI. It's a small, but difficult, step like almost every other advance in AI, and I admire the work done by those working in this field.

The problems of solving simple differential equations and symbolic integration at the first year calculus level are not really advanced math. Humans solve these problems with a relatively small bag of tricks that transform a symbolic integral into a simpler form. A program can do the same thing with an even more detailed database of transforms that can be attempted at each point in the search tree until a simple solution is reached.

The article claims that the new program can solve difficult integrals. This is interesting because hard to solve integrals are often associated with real physical phenomena. See for example the triple integrals of W. F. van Peype which arose while he was studying magnetism in different materials. These, relatively, plain looking definite integrals stumped some of the world's most famous mathematicians. See [1] and/or [2] for their interesting history.

[1] Paul J. Nahin, Inside Interesting Integrals, Springer, 2015. Section 6.5, The Watson/van Peype Triple Integrals.

[2] I.J. Zucker, 70+ Years of the Watson Integrals, http://www.inp.nsk.su/~silagadz/Watson_Integral.pdf

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