Abstract
As I mentioned in the Preface, in 1697 John Bernoulli evaluated the exotic-looking integral
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Notes
- 1.
This is not the way Bernoulli did his original evaluation, but rather is the modern way. The evaluation of ∫ 10 xm lnn(x)dx that I give in the solutions uses the gamma function, which was still in the future in Bernoulli’s day. Bernoulli used repeated integration by parts, which in fact is perfectly fine. The lack of a specialized tool doesn’t stop a genius!
- 2.
For a detailed discussion of Schuster’s integral and Hardy’s transform solution, see my book Dr. Euler’s Fabulous Formula, Princeton 2006, pp. 263–274.
- 3.
Since the volume we are integrating over is infinite in extent in both the t and u directions, the corner at the origin is the only corner of the bottom slab.
- 4.
G. N. Watson, “Three Triple Integrals,” Quarterly Journal of Mathematics, 1939, pp. 266–276.
- 5.
W. F. van Peype, “Zur Theorie der Magnetischen Anisotropic Kubischer Kristalle Beim Absoluten Nullpunkt,” Physica, June 1938, pp. 465–482. That is, van Peype was studying magnetic behavior in certain cubic crystalline lattice structures at very low temperatures (low means near absolute zero). The Watson/van Peype integrals turn-up not only in the physics of frozen magnetic crystals, but also in the pure mathematics of random walks. You can find a complete discussion of both the history and the mathematics of the integrals in I. J. Zucker, “70+ Years of the Watson Integrals,” Journal of Statistical Physics, November 2011, pp. 591–612.
- 6.
Integrals with the integrands \( \frac{1}{\sqrt{1-{\mathrm{k}}^2{ \sin}^2\left(\uptheta \right)}} \) and \( \sqrt{1-{\mathrm{k}}^2{ \sin}^2\left(\uptheta \right)} \) are elliptic integrals of the first and second kind, respectively (there is a third form, too). Such integrals occur in many important physical problems, such as the theory of the non-linear pendulum. As another example, the Italian mathematical physicist Galileo Galilei (1564–1642) studied the so-called “minimum descent time” problem, which involves an elliptic integral of the first kind, and its evaluation puzzled mathematicians for over a century. Eventually the French mathematician Adrien Marie Legendre (1752–1833) showed that the reason for the difficulty was that such integrals are entirely new functions, different from all other known functions. You can find more about Galileo’s problem, and the elliptic integral it encounters, in my book When Least is Best, Princeton 2007, pp. 200–210 and 347–351. A nice discussion of the non-linear pendulum, and the numerical evaluation of its elliptic integral, is in the paper by T. F. Zheng et al., “Teaching the Nonlinear Pendulum,” The Physics Teacher, April 1994, pp. 248–251.
- 7.
In this notation, the angle ϕ is measured from the positive x-axis and θ is measured from the positive z-axis. Some authors reverse this convention, but of course if one maintains consistency from start to finish everything comes out the same. The symbols are, after all, just squiggles of ink.
- 8.
I can only imagine what Watson’s words to his cat must have been when he reached this point in his work. Perhaps, maybe, they were something like this: “By Jove, Lord Fluffy, I’ve done it! Cracked the damn thing wide-open, just like when that egg-head Humpty-Dumpty fell off his bloody wall!”
- 9.
See M. G. Calkin and R. H. March, “The Dynamics of a Falling Chain. I,” American Journal of Physics, February 1989, pp. 154–157.
- 10.
The reason is that the falling part of the rope does so under the influence of not just gravity alone, but also from the non-zero tension in it that joins with gravity in pulling the rope down. To pursue this point here would take us too far away from the theme of the book but, if interested, it’s all worked out in the paper cited in note 9. The ‘faster than gravity’ prediction was experimentally confirmed, indirectly, in note 9 via tension measurements made during actual falls. In 1997 direct photographic evidence was published, and today you can find YouTube videos on the Web clearly showing ‘faster than gravity’ falls.
- 11.
Mathematicians are just as interested in this problem, and in related problems, as are physicists. Indeed, the study of falling ropes and chains was initiated by the British mathematician Arthur Cayley (1821–1895): see his note “On a Class of Dynamical Problems,” Proceedings of the Royal Society of London 1857, pp. 506–511, that opens with the words “There are a class of dynamical problems which, so far as I am aware, have not been considered in a general manner.” That’s certainly not the case today, with Cayley’s problem in particular still causing debate over when energy is (and isn’t) conserved: see Chun Wa Wong and Kosuke Yasui, “Falling Chains,” American Journal of Physics, June 2006, pp. 490–496. Take a look, too, at Challenge Problem 6.4.
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Nahin, P.J. (2015). Seven Not-So-Easy Integrals. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1277-3_6
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