Abstract
The immediate point of this opening section is to address the question of whether you will be able to understand the technical commentary in the book. To be blunt, do you know what an integral is? You can safely skip the next few paragraphs if this proves to be old hat, but perhaps it will be a useful over-view for some. It’s far less rigorous than a pure mathematician would like, and my intent is simply to define terminology.
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Notes
- 1.
For more on r(x), see E. Hairer and G. Wanner, Analysis by Its History, Springer 1996, p. 232, and William Dunham, The Calculus Gallery, Princeton 2005, pp. 108–112.
- 2.
Comment made after the presentation by E. C. Francis of his paper “Modern Theories of Integration,” The Mathematical Gazette, March 1926, pp. 72–77.
- 3.
In Hamming’s paper “Mathematics On a Distant Planet,” The American Mathematical Monthly, August-September 1998, pp. 640–650.
- 4.
See Bartle’s award-winning paper “Return to the Riemann Integral,” The American Mathematical Monthly, October 1996, pp. 625–632. He was professor of mathematics at the University of Illinois for many years, and then at Eastern Michigan University.
- 5.
Writing sin(1) means, of course, the sine of 1 radian = \( \frac{180^{{}^{\circ}}}{\uppi}={57.3}^{{}^{\circ}} \) (not of 1 degree).
- 6.
I am assuming that when you see \( {\displaystyle \int \frac{1}{{\mathrm{a}}^2+{\mathrm{x}}^2}\ \mathrm{dx}} \) you immediately recognize it as \( \frac{1}{\mathrm{a}}{ \tan}^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right) \). This is one of the few ‘fundamental’ indefinite integrals I’m going to assume you’ve seen previously from a first course of calculus. Others are: \( {\displaystyle \int \frac{1}{\mathrm{x}}\mathrm{dx}}= \ln \left(\mathrm{x}\right) \), ∫ ex dx = ex, \( {\displaystyle \int }{\mathrm{x}}^{\mathrm{n}}\ \mathrm{dx}=\frac{{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}\ \left(\mathrm{n}\ne -1\right) \), \( {\displaystyle \int}\frac{\mathrm{dx}}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}={ \sin}^{-1}\left(\mathrm{x}\right), \) and ∫ ln(x)dx = x ln(x) − x.
- 7.
This integral appeared in Feynman’s famous paper “Space-Time Approach to Quantum Electrodynamics,” Physical Review, September 15, 1949, pp. 769–789. Some historical discussion of the integral is in my book Number–Crunching, Princeton 2011, pp. xx–xxi.
- 8.
‘At random’ has the following meaning. If we look at any tiny patch of area dA in the interior of C1, a patch of any shape, then the probability a point is selected from that area patch is dA divided by the area of C1. We say that each of the three points is selected uniformly from the interior of C1.
- 9.
I think it almost intuitively obvious that the probability is scale-invariant (the same for any value of a), but just in case it isn’t obvious for you I’ll carry the radius of C1 along explicitly. At the end of our analysis you’ll see that the scale-setting parameter a has disappeared, proving my claim.
- 10.
The analysis I’ve just taken you through is the one given on pp. 817–818 of Edwards’ book that I mentioned in the Preface. The result of \( \frac{2\uppi}{15} \) is the answer derived by Edwards.
- 11.
This interval of estimates is a result of the code’s use of a random number generator (with the rand command)—every time we run the code we get a new estimate that is (slightly) different from the estimates produced by previous runs.
- 12.
Imagine that we have defined the maximum absolute value of a step, which will be our unit distance. Then, in one dimension (call it x) the particle moves, after each molecular hit, a distance randomly selected from the interval − 1 to +1. In a second, perpendicular direction (call it y) the particle moves, after each molecular hit, a distance randomly selected from the interval − 1 to +1. Figure 1.8.2 shows the combined result of these two independent motions for four particles, each for 1,000 hits (the four curves are MATLAB simulations).
- 13.
If you are curious about the details of such a derivation, you can find them in my book Mrs. Perkins’s Electric Quilt, Princeton University Press 2009, pp. 263–267.
- 14.
For a proof of this, see Mrs. Perkins’s, pp. 282–283.
- 15.
In the essay titled “A Different Box of Tools,” in Surely You’re Joking, Mr. Feynman!, W. W. Norton 1985, pp. 84–87.
- 16.
In the essay “Los Alamos from Below” in Surely You’re Joking, Mr. Feynman!, pp. 107–136.
- 17.
For a discussion of how this integral appears in a physics problem, see my book Mrs. Perkins’s Electric Quilt, Princeton 2009, pp. 2–3 (and also p. 4, for how to attack the challenge question—but try on your own before looking there or at the solutions).
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Nahin, P.J. (2015). Introduction. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1277-3_1
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