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ELLIPTIC INTEGRALS and SOME APPLICATIONS Jay Villanueva

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ELLIPTIC INTEGRALS and SOME APPLICATIONS Jay Villanueva ELLIPTIC INTEGRALS AND SOME APPLICATIONS Jay Villanueva Florida Memorial University 15800 NW 42nd Ave Miami, FL 33054 [email protected] 1. Introduction 1.1 How they arose – perimeter of an ellipse 1.2 Standard forms from Legendre, Jacobi, and Landen 2. Properties and examples 2.1 Evaluating integrals 2.2 Elliptic functions 3. Applications 3.1 Calculus 3.1.1 Perimeter of an ellipse 3.1.2 Arclength of a lemniscate 3.2 Physics 3.2.1 Finite-amplitude pendulum 3.2.2 Perihelion of Mercury 4. Conclusion References: 1. JW Armitage and WF Eberlein, 2006. Elliptic Functions. Cambridge University Press. 2. HM Kean and V Moll, 1997. Elliptic Curves. Cambridge University Press. 3. DF Lawden, 1989. Elliptic Functions and Applications. New York: Springer Verlag. 4. V Prasolov and Y Sovolyev, 1997. Elliptic Functions and Elliptic Integrals. Providence, RI: American Mathematical Society. 5. HE Rauch and A Lebowitz, 1973. Elliptic Functions, Theta Functions, and Riemann Surfaces. MD: Williams and Wilkins Co. 6. Schaum’s Outlines, 1996. Advanced Calculus. NY: McGraw-Hill. Elliptic integrals arose from the attempts to find the perimeter of an ellipse: 푥2 푦2 + = 1, 푥 = 푎 cos 휃, 푦 = 푏 sin 휃 푎2 푏2 푎 2휋 퐿 = 푑푠 = 2 √1 + 푦′2푑푥 = 1 + 푏2푐표푠2휃(−푎푠푛휃)푑휃, ∫ ∫−푎 ∫0 √ which last integral cannot be evaluated by elementary functions, viz., trigonometric, exponential, or logarithmic functions. The incomplete elliptic integral of the first kind is defined as 휙 푑휃 푢 = 퐹(푘, 휙) = ∫ , 0 < 푘 < 1, 0 √1−푘2푠푛2휃 where 휙 is the amplitude of 퐹(푘, 휙) or u, written 휙 = am 푢, and k is the modulus, 푘 = mod 푢. The integral is also called Legendre’s form for the elliptic integral of the first kind. If 휙 = 휋⁄2, the integral is called the complete integral of the first kind, denoted by 퐾(푘), or simply K. The incomplete elliptic integral of the second kind is defined by 휙 퐸(푘, 휙) = √1 − 푘2푠푛2휃 푑휃, 0 < 푘 < 1, ∫0 also called Legendre’s form for the elliptic integral of the second kind. If 휙 = 휋⁄2, the integral is called the complete elliptic integral of the second kind, denoted by 퐸(푘), or simply E. This is the form that arises in the determination of the length of arc of an ellipse. The incomplete elliptic integral of the third kind is defined by 휙 푑휃 퐻(푘, 푛, 휙) = ∫ , 0 < 푘 < 1, 푛 ≠ 0, 0 (1+푛푠푛2휃)√1−푘2푠푛2휃 also called Legendre’s form for the elliptic integral of the third kind. If the transformation 푣 = sin 휃 is made in the Legendre forms, we obtain the following integrals, with 푥 = sin 휙 푥 푑푣 퐹 (푘, 푥) = ∫ , 1 0 √(1−푣2)(1−푘2푣2) 푥 1−푘2푣2 퐸 (푘, 푥) = ∫ √ 푑푣, 1 0 1−푣2 푥 푑푣 퐻 (푘, 푛, 푥) = ∫ , 1 0 (1+푛푣2)√(1−푣2)(1−푘2푣2) called Jacobi’s forms for the elliptic integrals of the first, second, and the third kinds respectively. These are complete integrals if 푥 = 1. Using Landen’s transformation sin 2휙1 tan 휙 = or 푘 sin 휙 = sin(2휙1 − 휙), 푘+cos 2휙1 we can rewrite 2 2 휙1 푑휙1 2√푘 퐹(푘, 휙) = 퐹(푘1, 휙1) = ∫ , where 푘1 = , 푘 < 푘1 < 1. 1+푘 1+푘 0 2 2 1+푘 √1−푘1푠푛 휙1 By successive applications of the transformation, we obtain a sequence of moduli 2√푘푖−1 푘푛, 푛 = 1,2,3, …, 푘 = , lim푛→∞ 푘푛 = 1, 1+푘푖−1 from which we can write 푘 푘 푘 … 휙 푑휃 푘 푘 푘 … 휋 Φ 퐹(푘, 휙) = √ 1 2 3 ∫ = √ 1 2 3 ln 푡푎푛 ( + ) , Φ = lim 휙 . 푘 0 √1−푠푛2휃 푘 4 2 푛→∞ 푛 In practice, accurate results are obtained after only a few applications of the transformation. Many integrals are reducible to elliptic type. If 푅(푥, 푦) is a rational algebraic function of x and y, i.e., the quotient of two polynomials in x and y, then ∫ 푅(푥, 푦)푑푥 can be evaluated in terms of the usual elementary functions (algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic) if 푦 = √푎푥 + 푏 or 푦 = √푎푥2 + 푏푥 + 푐, with a,b,c constants. If 푦 = 푃(푥), with 푃(푥) a cubic or quartic polynomial, the integral R can be evaluated in terms of elliptic integrals of first, second, or third kinds, or for special cases in terms of elementary functions. If P is a polynomial of degree > 4, then R may be evaluated with the aid of hyper-elliptic functions. 2 푑푥 Example 1. ∫ . Let 푥 = 2 sin 휃. The integral becomes 0 √(4−푥2)(9−푥2) 휋/2 푑휃 1 휋/2 푑휃 1 2 휋 ∫ = ∫ = 퐹 ( , ) . 0 √9−4푠푛2휃 3 0 4 3 3 2 √1− 푠푛2휃 9 1 푑푥 Example 2. ∫ . Let 푥 = tan 휃. The integral becomes 0 √(1+푥2)(1+2푥2) 휋/4 푠푒푐2휃푑휃 휋/4 푑휃 휋/4 푑휃 1 휋/4 푑휃 ∫ = ∫ = ∫ = ∫ . 0 √1+푡푎푛2휃√1+2푡푎푛2휃 0 √푐표푠2휃+2푠푛2휃 0 √2−푐표푠2휃 √2 0 1 √1− 푐표푠2휃 2 Let 휙 = 휋⁄2 − 휃. The integral becomes 1 휋/2 푑휙 1 1 휋 1 휋 ∫ = {퐹 ( , ) − 퐹 ( , )}. √2 휋/4 1 √2 √2 2 √2 4 √1− 푠푛2휙 2 6 푑푥 Example 3. ∫ . Let 푢 = √푥 − 3 or 푥 = 3 + 푢2. The integral becomes 4 √(푥−1)(푥−2)(푥−3) √3 푑푢 2 ∫ . Let 푢 = tan 휃. The integral becomes 1 √(푢2+2)(푢2+1) 휋/3 푑휃 휋/3 푑휃 휋/3 푑휃 2 ∫ = 2 ∫ = √2 ∫ 휋/4 √2푐표푠2휃+푠푛2휃 휋/4 √2−푠푛2휃 휋/4 1 √1− 푠푛2휃 2 1 휋 1 휋 = √2 {퐹 ( , ) − 퐹 ( , )}. √2 3 √2 4 푑푥 In general, ∫ , where P is a 3rd or 4th degree polynomial, can be evaluated by elliptic √푃 integrals. The elliptic functions are defined via the elliptic integrals. The upper limit x in the Jacobi form of the elliptic integral of the first kind is related to the upper limit 휙 in the Legendre form by 푥 = sin 휙. Since 휙 = am 푢, it follows that 푥 = sin(am 푢). We define the elliptic functions 푥 = sin(am 푢) = 푠푛 푢 √1 − 푥2 = cos(푎푚 푢) = 푐푛 푢 √1 − 푘2푥2 = √1 − 푘2푠푛2푢 = 푑푛 푢 푥 푠푛 푢 = = 푡푛 푢. √1−푥2 푐푛 푢 It is also possible to define inverse elliptic functions. For example, from 푥 = 푠푛 푢, we define 푢 = 푠푛−1푥, or 푢 = 푠푛−1(푥, 푘) = 푠푛−1푥, mod 푘, to show the dependence of u on k. These functions have many important properties analogous to those of trigonometric functions. One special property is the fact that these functions are doubly periodic, one period is real, the other is complex. If 휋/2 푑휃 퐾 = ∫ , 0 √1−푘2푠푛2휃 then 푠푛(푢 + 4퐾) = 푠푛 푢 푐푛(푢 + 4퐾) = 푐푛 푢 푑푛(푢 + 2퐾) = 푑푛 푢 푡푛(푢 + 2퐾) = 푡푛 푢. These functions also have other periods, which are complex. If 휋/2 푑휃 퐾′ = ∫ , 푘′ = √1 − 푘2, 0 √1−푘′2푠푛2휃 Then 푠푛 푢 has periods 4퐾 and 2퐾′; 푐푛 푢 has periods 4퐾 and 2퐾 + 2퐾′; and dn u has periods 2퐾 and 4퐾′. For this reason, the elliptic functions are called doubly-periodic. Application 1. Perimeter of an ellipse. The ellipse 푥 = 푎 cos 휃, 푦 = 푏 sin 휃, 푎 > 푏 > 0, has length 휋/2 휋/2 퐿 = 4 √푑푥2 + 푑푦2 = 4 푎2푐표푠2휃 + 푏2푠푛2휃푑휃 ∫0 ∫0 √ 휋/2 휋/2 = 4 푎2 − (푎2 − 푏2)푠푛2휃푑휃 = 4푎 √1 − 푒2푠푛2휃푑휃, ∫0 √ ∫0 where 푒2 = (푎2 − 푏2)⁄푎2 = 푐2/푎2 is the square of the eccentricity of the ellipse. 휋 The result can be written as 퐿 = 4푎퐸 (푒, ) = 4푎퐸(푒). For the special case of a circle, 2 푎 = 푏 = 푟, i.e., 푒 = 0, and 퐸(0) = 휋/2, and we recover the circumference of a circle: 퐿 = 2휋푟. The term elliptic integral was coined by Count Fagnano (1682 – 1766) in 1750. He discovered that the arclength of the lemniscate can be expressed in terms of an elliptic integral of the first kind. Application 2. Arclength of a lemniscate. The lemniscate is the figure 8 curve: (푥2 + 푦2)2 = 푎2(푥2 − 푦2)2, or in polar form 푟2 = 푎2 cos 2휃. From 푑푠2 = 푑푥2 + 푑푦2 = 푑푟2 + 푟2푑휃2, 휋/4 1 1 퐿 = 4 ∫ 푑푠 = 4푎 ∫ 푑푟, 푟 = tan 휃 ⇒ 푟=0 0 √1−푟4 1 1 휋/4 푠푒푐2휃푑휃 휋/4 푑휃 ∫ 푑푟 = ∫ = ∫ , (cos 2휃 = 푐표푠2푢) ⇒ 0 √1−푟4 0 푠푒푐휃√1−푡푎푛2휃 0 √cos 2휃 휋/2 푑푢 1 휋/2 푑푢 1 1 = ∫ = ∫ = ∙ 퐾 ( ) . 0 √2−푠푛2푢 √2 0 1 √2 √2 √1− 푠푛2푢 2 1 1 Thus, 퐿 = 4푎 ∙ 퐾 ( ) = 푎 ∙ 2√2(1.85407) = 푎(5.244102). √2 √2 (Historical note: The rectification of the lemniscate was first done by Fagnano in 1718. The lemniscatus, L.‘decorated by ribbons’, was first studied in astronomy in 1680 by Cassini, known as the ovals of Cassini (Figure 3), but his book was published in 1749, many years after his death. The curves were popularized by the Bernoulli brothers in 1694.) Cassini considered more general forms of the lemniscate for whose points the products of the distances to two foci is a constant: 2 푑1푑2 = 푏 , 푎2 푏4 = 푟4 + − 푟2푎2 cos 2휃. 4 푎 When 푏 = , centered at the origin, we get the ribbon-shaped curve. √2 Application 3. Finite-amplitude pendulum. The equation of motion is: 푑푝 ̇ 푚푙휃̈ = −푚푔 sin 휃. Let 푝 = 휃̇ → 푝 = − sin 휃 푑휃 푙 푝2 ⇒ = cos 휃 + 퐶. 2 푙 푑휃 2 I.C.: At 푡 = 0: 휃 = 휃 , 휃̇ = 0 ⇒ = −√ √cos 휃 − cos 휃 .
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