Abstract
To start this chapter, here’s a very simple illustration of the power series technique for the calculation of integrals, in this case giving us what is called Catalan’s constant (mentioned in the Preface) and written as G = 0.9159655 … . The power series expansion of tan− 1(x), for |x| < 1, is our starting point, and it can be found as follows. The key idea is to write
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Notes
- 1.
I mentioned this sum earlier, in the Introduction (Sect. 1.3). Later in this book, in Chap. 7, I’ll show you a beautiful way—using integrals, of course!—to derive this famous and very important result. What I’ll show you then is not the way Euler did it, but it has the distinct virtue of being perfectly correct while Euler’s original approach (while that of a genius—for details, see my book An Imaginary Tale, Princeton 2010, pp. 148–9) is open to some serious mathematical concerns.
- 2.
This reversal is an example of a step where a mathematician would feel obligated to first show uniform convergence before continuing. I, on the other hand, with a complete lack of shame, will just plow ahead and do the reversal and then, once I have the ‘answer,’ will ask quad what it ‘thinks.’
- 3.
See my Dr. Euler’s Fabulous Formula, Princeton 2011, p. 149 for the derivation of this result using Fourier series.
- 4.
For s = 1, ζ(1) is just the harmonic series, which has been known for centuries before Euler’s day to diverge.
- 5.
A technically sophisticated yet quite readable treatment of ‘all about γ,’ at the level of this book, is Julian Havil’s Gamma, Princeton University Press 2003. The constant is also sometimes called the Euler-Mascheroni constant, to give some recognition to the Italian mathematician Lorenzo Mascheroni (1750-1800) who, in 1790, calculated γ to 32 decimal places (but, alas, not without error). As I write, γ has been machine-calculated to literally billions of decimal places, with the first few digits being 0.5772156649 …. . Unlike π or e which are known to be irrational (transcendental, in fact), the rationality (or not) of γ is unknown. There isn’t a mathematician on the planet who doesn’t believe γ is irrational, but there is no known proof of that belief.
- 6.
I won’t pursue the derivation of \( \frac{\Gamma^{\prime}\left(\mathrm{z}\right)}{\Gamma \left(\mathrm{z}\right)} \) , but if you’re interested you can find a nice presentation in Havil’s book (see note 5, pp. 55–58). We’ll use the digamma function in the final calculation of this chapter, the derivation of (5.4.11). See also Challenge Problem 5.10.
- 7.
At the upper limit the integrand does become the indeterminate \( \frac{0}{0} \), but we can use L’Hospital’s rule to compute the perfectly respectable \( { \lim}_{\mathrm{x}\to 1}\frac{1-{\mathrm{x}}^{\mathrm{m}}}{1-{\mathrm{x}}^{\mathrm{n}}}={ \lim}_{\mathrm{x}\to 1}\frac{-\mathrm{m}\ {\mathrm{x}}^{\mathrm{m}-1}}{-\mathrm{n}\ {\mathrm{x}}^{\mathrm{n}-1}}=\frac{\mathrm{m}}{\mathrm{n}} \).
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Nahin, P.J. (2015). Using Power Series to Evaluate Integrals. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1277-3_5
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