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(Almost) Impossible Integrals, Sums, and Series

Cornel loan Valean
Publication Date: 
Number of Pages: 
Problem Books in Mathematics
Problem Book
[Reviewed by
Warren Johnson
, on
If you like to solve the problems on definite integrals in The American Mathematical Monthly, or would like to be better at solving them, then you should look at the book under review, which I will refer to as Impossible.  It is also the state of the art for a very particular kind of infinite series.
Two related books are Irresistible Integrals, by George Boros and Victor H. Moll, which I will refer to as Irresistible, and Inside Interesting Integralsby Paul J. Nahin, which I will call Interesting.  Valean's correspondence with Nahin is partly responsible for Impossible, and Nahin's Foreword compares Valean to Boros, the "person who could do any integral."   Valean does not appear to be familiar with Joseph Edwards's A Treatise on the Integral Calculus, which might have helped him in several places.  Other older books worth mentioning are Greenhill's A Chapter in the Integral Calculus and Bromwich's Elementary Integrals.  Hardy's A Course of Pure Mathematics also has a lot of valuable material on techniques of integration, especially in the exercises.    
Like Irresistible and Interesting, Impossible is concerned only with definite integrals, whereas Bromwich and Greenhill are primarily interested in indefinite integrals and Edwards treats both.  Like Irresistible, Impossible uses only real variable methods (except for an occasional \(e^{i\theta}=\cos\theta+i\sin\theta\)), whereas Nahin permits himself contour integration in one chapter of Interesting.  Unlike Irresistible, which one can teach a fine course from (if anyone can be found to take it), Impossible is an eccentric mixture of problem book and research monograph.  It could be a supplementary text for a course based on Irresistible, but I would rather use Interesting or Edwards for this purpose.  Impossible comprises two sets of three chapters, with each chapter comprising 60 sections.  The first and fourth chapters are problems, the former on integrals and the latter on series; the second and fifth chapters are hints, and the third and sixth chapters are solutions.  This explains why chapters 1-3 have the same number of sections, and why chapters 4-6 have the same number.  That all six have the same number is presumably a whim of the author.  The problems occupy \(37+36\) pages, the hints
\(15+14\), and the solutions \(219+206\).
The biggest problem with Impossible is that Valean does not write nearly as well as Nahin, nor even as well as Boros and Moll, to say nothing of Hardy.   He does not understand the definite article in English, consistently writing "the integration by parts", for example.  It is good to see someone with an unconventional background doing mathematics of this quality (Valean was trained as an accountant), but I think his lack of teaching experience hurts the exposition.   He does not distinguish enough between a solution and a good solution.  
In the body of his solutions he usually writes the smallest amount of prose that he can get away with, but the section titles are expansive; for example, that of 1.51, 2.51, and 3.51 is "Playing with a Resistant Classical Integral Family to the Real Methods that Responds to the Tricks Involving the Use of the Cauchy-Schlömilch Transformation" (yes, he really wrote all of that six times.)  
Impossible is a very personal book, with an idiosyncratic selection of material.  Trigonometric integrals are infrequent. Surprisingly absent is 
\( \int_{0}^{\infty} \frac{\sin x}{x} dx = \frac{\pi}{2}, \)
which is also a curious omission from Irresistible, although a related result is stated without proof on p. 136.  One of the prettiest calculations in Impossible is 3.56, "A Double Integral Hiding a Beautiful Idea About the Symmetry and (Possibly) an Unexpected Closed-Form", which is
\( M=\int_{0}^{\frac{\pi}{4}} \int_{0}^{\frac{\pi}{4}} \arctan \left( \cos x \cot y \right) \sec^{3} x \tan^{2} y \sec^{2} y \; dx \; dy. \)
Valean first reverses the order to \(dy\,dx\) and then, in effect, replaces \(x\) by \(\theta\)
and substitutes \(r=\tan y\sec\theta\), resulting in
\( M=\int_{0}^{\frac{\pi}{4}} \int_{0}^{\sec \theta} r^{2} \arctan \left( \frac{1}{r} \right) \; dr \; d\theta . \)
He has \(t\) and \(x\) for \(r\) and \(\theta\) at this point, but he now conceives of the variables as polar coordinates and changes them to Cartesian.  Since \(r=\sec\theta\) is the line \(x=1\) and \(\theta=\frac{\pi}4\) is \(y=x\), this gives
\( M=\int_{0}^{1} \int_{0}^{x} \sqrt{x^{2}+y^{2}} \arctan \left( \frac{1}{\sqrt{x^{2}+y^{2}}} \right) \; dy \; dx, \)
which by symmetry is
\( M=\frac{1}{2} \int_{0}^{1} \int_{0}^{1} \sqrt{x^{2}+y^{2}} \arctan \left( \frac{1}{\sqrt{x^{2}+y^{2}}} \right) \; dy \; dx.\)
He then observes without comment that
\( \int_{0}^{1} \frac{x^{2}+y^{2}}{x^{2}+y^{2}+z^{2}} \; dz = \sqrt{x^{2}+y^{2}} \arctan \left( \frac{1}{\sqrt{x^{2}+y^{2}}} \right) , \)
which may have been the starting point for the construction of the problem.  Hence
\( M=\frac{1}{2} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{x^{2}+y^{2}}{x^{2}+y^{2}+z^{2}} \; dz \; dy \l dx \)
and another symmetry argument finishes the job:  the numerator could just as well be \(x^2+z^2\) or \(y^2+z^2\), and adding the three possibilities together we get 
\( 3M=\frac{1}{2} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{2(x^{2}+y^{2}+z^{2})}{x^{2}+y^{2}+z^{2}} \; dz \; dy \; dx = 1, \)
so \( M=1/3 \).  
In this step Valean writes much more than I did, unlike the previous steps where he writes less.  This is a beautiful example, and I may use it the next time I teach multivariable calculus.
I don't know why "sums" and "series" are both in the title.  At one point I wondered whether this project should really have been two books, one on integrals and one on series, but there is enough overlap both in results and in methods that I think it is right to have them under one cover. 
Valean writes "For the readers I have the last words of the Preface:  Let's start the journey in the fascinating world of integrals, sums, and series and have much fun!"   He clearly enjoys doing these calculations and displaying his considerable virtuosity, but he doesn't present them well enough for the reader to enjoy them.  I am glad to have this book, but I am afraid its appeal may be rather limited.  It would have benefited from some editing.
Warren Johnson is Associate Professor of Mathematics at Connecticut College.  His first love in mathematics was techniques of integration.

See the publisher's web page.