Abstract
The continuing theme of this chapter is the development and use of the technique of differentiating an integral (popularly known as ‘Feynman’s trick’). Illustrative examples include some historically important integrals (the Gaussian probability integral, Dirichlet’s discontinuous integral, and Dini’s integral). The idea of recursion, introduced in the previous chapter, appears again. The trick of inverting Feynman’s trick by integrating the integral of interest to make a double integral and then reversing the order of integration is introduced. The Cauchy-Schlӧmilch transformation is stated, derived, and used to evaluate some interesting variations of the probability integral.
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Notes
- 1.
The probability integral is most commonly evaluated in textbooks with the trick of converting it to a double integral in polar coordinates (see, for example, my books An Imaginary Tale: the story of \( \sqrt{-1} \), Princeton 2012, pp. 177–178, and Mrs. Perkins’s Electric Quilt, Princeton 2009, pp. 282–283), and the use of Leibniz’s formula that I’m going to show you here is uncommon. It is such an important integral that at the end of this chapter we will return to it with some additional analyses.
- 2.
The reason we write a partial derivative inside the integral and a total derivative outside the integral is that the integrand is a function of two variables (t and y) while the integral itself is a function only of t (we’ve ‘integrated out’ the y dependency).
- 3.
After the American physicist J. W. Gibbs (1839–1903), who was the wrong person to have his name attached to the wiggles. For the history of this, see my book Dr. Euler’s Fabulous Formula, Princeton 2011, pp. 163–173.
- 4.
The integral in (3.4.8) occurs in a paper by R. M. Dimeo, “Fourier Transform Solution to the Semi-Infinite Resistance Ladder,” American Journal of Physics, July 2000, pp. 669–670, where it is done by contour integration (see the comments immediately following (8.8.11)). Our derivation here shows contour integration is actually not necessary, although Professor Dimeo’s comment that contour integration is “well within the abilities of the undergraduate physics major,” is in agreement with the philosophical position I take in the original Preface. Notice that we can use (3.4.8) to derive all sorts of new integrals by differentiation. For example, suppose b = −1, and so we have \( {\int}_0^{\uppi}\frac{1}{\mathrm{a}-\cos \left(\mathrm{x}\right)}\mathrm{dx}=\frac{\uppi}{\sqrt{{\mathrm{a}}^2-1}} \). Then, differentiating both sides with respect to a, we get \( {\int}_0^{\uppi}\frac{1}{{\left[\mathrm{a}-\cos \left(\mathrm{x}\right)\right]}^2}\mathrm{dx}=\frac{\uppi \mathrm{a}}{{\left({\mathrm{a}}^2-1\right)}^{3/2}} \). If, for example, a = 5, this new integral is equal to \( \frac{5\uppi}{24^{3/2}}=\frac{5\uppi}{48\sqrt{6}}=0.1335989\dots \). To check, we see that integral(@(x)1./((5-cos(x)).^2),0,pi) = 0.1335989… .
- 5.
For n = 1, the recursion gives I1 in terms of I0, where \( {\mathrm{I}}_0={\int}_0^{\uppi /2}\mathrm{dx}=\frac{\uppi}{2} \) with no dependency on either a or b. That is, \( \frac{\partial {\mathrm{I}}_0}{\mathrm{\partial a}}=\frac{\partial {\mathrm{I}}_0}{\mathrm{\partial b}}=0 \) and the recursion becomes the useless, indeterminate \( {\mathrm{I}}_1=\frac{0}{0} \) I0.
- 6.
Uhler was a pioneer in heroic numerical calculation, made all the more impressive in that he worked in the pre-electronic computer days. His major tool was a good set of log tables. I describe his 1921 calculation of, to well over 100 decimal digits, the value of \( {\left(\sqrt{-1}\right)}^{\sqrt{-1}}={\mathrm{e}}^{-\frac{\uppi}{2}} \) in my book An Imaginary Tale: the story \( \sqrt{-1} \), Princeton 2010, pp. 235–237. Why did Uhler perform such a monstrous calculation? For the same reason kids play with mud—he thought it was fun!
- 7.
Waldemar Klobus, “Motion On a Vertical Loop with Friction,” American Journal of Physics, September 2011, pp. 913–918.
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Nahin, P.J. (2020). Feynman’s Favorite Trick. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-43788-6_3
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