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Elliptic Integrals and Integration by Substitution Janet Heine Barnett Colorado State University-Pueblo, [email protected]

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Elliptic Integrals and Integration by Substitution Janet Heine Barnett Colorado State University-Pueblo, Janet.Barnett@Csupueblo.Edu Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Calculus Mathematics via Primary Historical Sources (TRIUMPHS) Spring 2019 Gaussian Guesswork: Elliptic Integrals and Integration by Substitution Janet Heine Barnett Colorado State University-Pueblo, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_calculus Click here to let us know how access to this document benefits oy u. Recommended Citation Barnett, Janet Heine, "Gaussian Guesswork: Elliptic Integrals and Integration by Substitution" (2019). Calculus. 8. https://digitalcommons.ursinus.edu/triumphs_calculus/8 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Calculus by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. Gaussian Guesswork: Elliptic Integrals and Integration by Substitution Janet Heine Barnett∗ August 18, 2019 Just prior to his 19th birthday, the mathematical genius Carl Friedrich Gauss (1777{1855) began a \mathematical diary" in which he recorded his mathematical discoveries for nearly 20 years. Among these discoveries is the existence of a beautiful relationship between three particular numbers: • the ratio of the circumference of a circle to its diameter, or π; Z 1 dx • a specific value of a certain (elliptic1) integral, which Gauss denoted2 by $ = 2 p ; and 4 0 1 − x p p • a number called \the arithmetic-geometric mean" of 2 and 1, which he denoted as µ( 2; 1). Like many of his discoveries, Gauss uncovered this particular relationship through a combination of the use of analogy and the examination of computational data, a practice that historian Adrian Rice calls \Gaussian Guesswork" in his Math Horizons article subtitled \Why 1:19814023473559220744 ::: is such a beautiful number" [Rice, November 2009]. This mini-project is one of a set of four mini-projects, based on excerpts from Gauss' mathematical diary [Gauss, 2005] and related manuscripts, that looks at the power of Gaussian guesswork via the story of his discovery of this beautiful relationship. In this mini-project, we focus especially on how Gauss' discovery led him to the creation of a powerful new technique for evaluating certain types of integrals, such as the one that defines the number $. R 1 Why study the integral 1 p dx ? 0 1−x4 RIf you have studied the family of curves known as lemniscates, then you may recall that the integral 1 p dx gives us the arc length of one-fourth of the lemniscate of equation (x2 + y2)2 = x2 − y2. Interest 0 1−x4 in the lemniscate and its arc length dates back to the earliest days of calculus, when the two Bernoulli brothers (Jacob and Johann) studied them in the late 1690's. It was Jacob Bernoulli who named the curve lemniscus, a Latin word for a type of a decorative ribbon shaped somewhat like the current symbol 1 for infinity. (See Figure 1.) From his notebooks, we know that Gauss himself studied this integral and the ∗Department of Mathematics and Physics, Colorado State University - Pueblo, Pueblo, CO 81001 - 4901; [email protected]. R 1 x p 1 ≥ An integral of the form 0 1−tn dt is called an elliptic integral for n = 3 and n = 4; for n 5, it is called hyperelliptic. This terminology is historically related to the occurrence of this form in connection to the arc length of ellipses and other curves that naturally arise in astronomy and physics. 2The symbol `$' that Gauss used to denote this specific value is called \varpi;" it is a variant of the Greek letter π. 1 lemniscate through the works of Euler, which he began to read by the age of 20. Gauss made note of these studies in his 51st diary entry,3 where he wrote: 1111111111111111111111111111111111111111 Z I have begun to examine thoroughly the elastic4 lemniscatic curve depending on (1 − x4)−1=2dx. January 8, 1797 1111111111111111111111111111111111111111 Figure 1: Sketch of the lemniscate from Gauss's notebook, from [Gauss, 1876] Naturally, Gauss was interested in evaluating the integral for the arc length of the lemniscate. His first idea for doing this is one that perhaps has also occurred to you, based on your experience with trigonometric substitution:5 1111111111111111111111111111111111111111 Z Z sin 1 1 : : : : : : xx = θ p dθ = 2 p dx cos sin θ 1 − x4 . January 7, 1797 1111111111111111111111111111111111111111 Gauss' notation is a bit cryptic, as he seemed to be describing two different substitutions that would lead him to the result indicated. In the following task, you will verify just one of these two substitutions. R Task 1 In this task, we consider the integral p 1 dx. 1−x4 (a) Consider the substitution x2 = sin θ. Explain why this is a natural substitution to try here. (b) Now verify that the substitution from part (a) gives the integral result that Gauss noted in his diary entry of January 7, 1797. What strategies might we use in order to evaluate the resulting dθ integral? What prevents these strategies from going through? 3A German version of Gauss' Diary appears in [Gauss, 2005]. In this project, we use the English translation by Jeremy Grey that appears in [Dunnington, 2004, pp. 469-484]. 4Originally, Gauss wrote `elastic' here, only to cross it out at some later unknown date when he instead wrote in `lemniscatic.' See the author's primary source project Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve for information about the elastic curve and it relationship to the lemniscate, as well as their role in the early history of calculus. 5The symbols used by Gauss in this and other diary entries have been changed slightly in order to maintain notational consistency throughout the project, which uses excerpts from several different works by Gauss. 2 Interestingly, there is a slight variation on the substitution from Task 1 which allowed Gauss to even- tually succeed in evaluating the lemniscate arc length integral. In the next task, we will use this other substitution to transform this integral into a new form | a form that may not seem promising at first glance. In the subsequent sections of this project, we will then work through Gauss' treatment of the transformed integral to see how he was eventually able to evaluate it. Z Z 1 dx π=2 dθ Task 2 (a) Let x = sin θ. Show p = p . 4 0 1 − x 0 1 + sin2 θ Z Z π=2 dθ π=2 dθ (b) Now re-write p as an integral of the form p . 0 1 + sin2 θ 0 m2 cos2 θ + n2 sin2 θ State the values of m and n clearly. 2 The Arithmetic-Geometric Mean ZIn the next section of this project, we will see how Gauss finally managed to evaluate integrals of the form π=2 dθ p . In this section, we first take a look at the mathematical tool that would 0 m2 cos2 θ + n2 sin2 θ eventually allow him to tackle this tricky integral. This tool | a type of average called the arithmetic- geometric mean | was discovered by Gauss independently of his work on the lemniscate, but also quite early in his mathematical studies. Although his Nachlass contains extensive notes about the arithmetic- geometric mean and its properties, it was mentioned only once in his published works, in an important astronomical paper on the gravitational attraction of planets [Gauss, 1818]. Here is what Gauss had to say about the arithmetic-geometric mean in that paper6: 1111111111111111111111111111111111111111 Let m; n be two positive quantities, and set7 1 p m = (m + n); n = mn 1 2 1 so that m1, n1 represent the arithmetic mean and the geometric mean, respectively, of m and n. The geometric mean will always be taken to be positive. Similarly set p m = 1 (m + n ); n = m n 2 2 1 1 2 p 1 1 1 m3 = 2 (m2 + n2); n3 = m2n2 and so on, by which manner [are obtained] the sequences m; m1; m2; m3; etc., and n; n1; n2; n3; etc., converging rapidly to a common limit, which we denote µ, and call simply the arithmetic-geometric mean between m and n. 1111111111111111111111111111111111111111 6The excerpts from Determinatio Attratctionis, quam in punctum quodvis positionis datae exerceret planeta, si eius massa per totam orbitam ratione temoris, quo singulae partes descibuntur, uniformiter esset dispertita used in this project are taken from pages 352{353 of [Gauss, 1818]. The Latin translations of these excerpts were done by George W. Heine III, Math and Maps ([email protected]). 7Gauss himself used prime notation (i.e., m0; m00; m000 ) to denote the terms of the sequence. In this project, we instead use indexed notation (i.e., m1; m2; m3 ) in keeping with current notational conventions. To fully adapt Gauss' notation to that used today, we could also write m0 = m and n0 = n. 3 Notice that the value of the arithmetic-geometric mean µ depends on the starting values that are used for m and n; we will use the function notation µ(m; n) to remind us of this fact.8 Although Gauss did not give any examples of how to compute arithmetic-geometric mean in his 1818 astronomy paper, here is one of the examples9 that he worked out much earlier in one of his unpublished Nachlass papers, [Gauss, 1799]. 1111111111111111111111111111111111111111 Example 1: m = 1, n = 0:2 m = 1:00000 00000 00000 00000 0 n = 0:20000 00000 00000 00000 0 m1 = 0:60000 00000 00000 00000 0 n1 = 0:44721 35954 99957 93928 2 m2 = 0:52360 67977 49978 99964 1 n2 = 0:51800 40128 22268 36005 0 m3 = 0:52080 54052 86123 95414 3 n3 = 0:52080 78709 39876 24344 0 m4 = 0:52080 16381 06187 n4 = 0:52080 16381 06187 rd Here m5; n5 differ in the 23 decimal place.
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