Abstract
In this chapter we present a small sample of some of the most beautiful formulas in the world.
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Notes
- 1.
“Formal” in mathematics usually refers to “having the form or appearance without the substance or essence,” which is the fifth entry for “formal” in Webster’s 1828 dictionary. This is very different from the common use of “formal”: “according to form; agreeable to established mode; regular; methodical,” which is the first entry in Webster’s 1828 dictionary. Elaborating on the mathematical use of “formal,” it means something like “a symbolic manipulation or expression presented without paying attention to correctness”.
- 2.
Here, Euler set \(y = \sin s\).
- 3.
Instead of writing, e.g., \(1 \cdot 2 \cdot 3\), today we would write this as 3!. However, the factorial symbol wasn’t invented until 1808 [44, p. 341], by Christian Kramp (1760–1826), more than 70 years after De summis serierum reciprocarum was read in the St. Petersburg Academy.
- 4.
Here, Euler uses p for \(\pi \). The notation \(\pi \) for the ratio of the length of a circle to its diameter was introduced in 1706 by William Jones (1675–1749), and around 1736, a year after Euler published De summis serierum reciprocarum, Euler seems to have adopted the notation \(\pi \).
- 5.
Here, ss means \(s^2\).
- 6.
It took \(\approx 5\) years to find this passage! The breakthrough came thanks to Emanuele Delucchi, who contacted his sister Rachele Delucchi, who then found Mengoli’s book in the library of ETH Zurich, and thanks to Emanuele Delucchi for translating the original Latin.
- 7.
Explicitly, \(C_{k} = B_{2k}/(2k)!\), but this formula is not needed.
- 8.
Actually, this works in reverse: We can just as well take Euler’s formula as given, and then derive Gregory–Leibniz–Madhava’s formula!
- 9.
Alternatively, one can prove that \(\frac{1}{(2m + 1)(2n + 1)} = \frac{1}{2(m - n)(2n + 1)} + \frac{1}{2(n - m)(2m + 1)}\) and use this decomposition in what follows. However, the decomposition of \(\frac{1}{(2m + 1)(2n + 1)}\) as presented might be helpful if you do “Williams’s other formula” in Problem 4.
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© 2017 Paul Loya
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Loya, P. (2017). Some of the Most Beautiful Formulas in the World I–III. In: Amazing and Aesthetic Aspects of Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6795-7_5
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