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Elliptic Integrals, Elliptic Functions and Theta Functions Reading Problems Outline Background ...................................................................2 Theory .........................................................................6 Elliptic Integral .........................................................6 Elliptic Function .......................................................14 Theta Function .........................................................16 Assigned Problems ..........................................................17 References ....................................................................20 1 Background This chapter deals with the Legendre elliptic integrals, the Theta functions and the Jaco- bian elliptic functions. These elliptic integrals and functions find many applications in thetheory of numbers, algebra, geometry, linear and non-linear ordinary and partial differentialequations, dynamics, mechanics, electrostatics, conduction and field theory. An elliptic integral is any integral of the general form f(x)=/integraldisplayA(x)+B(x) C(x)+D(x)/radicalbig S(x)dx where A(x),B(x),C(x)andD(x)are polynomials in xandS(x)is a polynomial of degree 3 or 4. Elliptic integrals can be viewed as generalizations of the inverse trigonometricfunctions. Within the scope of this course we will examine elliptic integrals of the first andsecond kind which take the following forms: First Kind If we let the modulus ksatisfy 0≤k2<1(this is sometimes written in terms of the parameter m≡k2or modular angle α≡sin−1k). The incomplete elliptic integral of the first kind is written as F(φ, k)=/integraldisplaysinφ 0dt/radicalbig (1−t2)(1−k2t2),0≤k2≤1and 0≤sinφ≤1 if we let t=sinθanddt=cosθd θ=√1−t2dθ,then F(φ, k)=/integraldisplayφ 0dθ/radicalbig 1−k2sin2θ,0≤k2≤1and 0≤φ≤π/2 This is referred to as the incomplete Legendre elliptic integral. The complete elliptic integral can be obtained by setting the upper bound of the integral to its maximum range, i.e.sinφ=1orφ=π/2to give K(k)=/integraldisplay 1 0dt/radicalbig (1−t2)(1−k2t2) =/integraldisplayπ/2 0dθ/radicalbig 1−k2sin2θ 2 Second Kind E(φ, k)=/integraldisplaysinφ 0√1−k2t2 √1−t2dt =/integraldisplayφ 0/radicalbig 1−k2sin2θd θ Similarly, the complete elliptic integral can be obtained by setting the upper bound of inte- gration to the maximum value to get E(k)=/integraldisplay1 0√1−k2t2 √1−t2dt =/integraldisplayπ/2 0/radicalbig 1−k2sin2td t Another very useful class of functions can be obtained by inverting the elliptic integrals. As an example of the Jacobian elliptic function snwecan write u(x=sinφ, k)=F(φ, k)=/integraldisplaysinφ 0dt/radicalbig (1−t2)(1−k2t2) If we wish to find the inverse of the elliptic integral x=sinφ=sn(u, k) or u=/integraldisplaysn 0dt/radicalbig (1−t2)(1−k2t2) While there are 12 different types of Jacobian elliptic functions based on the number of poles and the upper limit on the elliptic integral, the three most popular are the copolar trio of sineamplitude, sn(u, k),cosine amplitude, cn(u, k)and the delta amplitude elliptic function, dn(u, k)where 3 sn2+cn2=1 and k2sn2+dn2=1 Historical Perspective The first reported study of elliptical integrals was in 1655 when John Wallis began to study the arc length of an ellipse. Both John Wallis (1616-1703) and Isaac Newton (1643-1727)published an infinite series expansion for the arc length of the ellipse. But it was not until thelate 1700’s that Legendre began to use elliptic functions for problems such as the movementof a simple pendulum and the deflection of a thin elastic bar that these types of problemscould be defined in terms of simple functions. Adrien-Marie Legendre (1752-1833), a French mathematician, is remembered mainly for the Legendre symbol and Legendre functions which bear his name but he spent more than fortyyears of his life working on elliptic functions, including the classification of elliptic integrals. His first published writings on elliptic integrals consisted of two papers in the Memoires del’Acadmie Francaise in 1786 based on elliptic arcs. In 1792 he presented to the Acadmie amemoir on elliptic transcendents. Legendre’s major work on elliptic functions appeared in 3 volumes 5in 1811-1816. In the first volume Legendre introduced basic properties of elliptic integrals as well as propertiesfor beta and gamma functions. More results on beta and gamma functions appeared in thesecond volume together with applications of his results to mechanics, the rotation of theEarth, the attraction of ellipsoids and other problems. The third volume contained the veryelaborate and now well-known tables of elliptic integrals which were calculated by Legendrehimself, with an account of the mode of their construction. He then repeated much of thiswork again in a three volume set 6in 1825-1830. Despite forty years of dedication to elliptic functions, Legendre’s work went essentially unno- ticed by his contemporaries until 1827 when two young and as yet unknown mathematiciansAbel and Jacobi placed the subject on a new basis, and revolutionized it completely. In 1825, the Norwegian government funded Abel on a scholarly visit to France and Germany. Abel then traveled to Paris, where he gave an important paper revealing the double period-icity of the elliptic functions. Among his other accomplishments, Abel wrote a monumentalwork on elliptic functions 7which unfortunately was not discovered until after his death. Jacobi wrote the classic treatise8on elliptic functions, of great importance in mathematical physics, because of the need to integrate second order kinetic energy equations. The motion 5Exercises du Calcul Intgral 6Trait des Fonctions Elliptiques 7Abel, N.H. “Recherches sur les fonctions elliptiques.” J. reine angew. Math. 3, 160-190, 1828. 8Jacobi, C.G.J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntraeger, 1829. 4 equations in rotational form are integrable only for the three cases of the pendulum, the symmetric top in a gravitational field, and a freely spinning body, wherein solutions are interms of elliptic functions. Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the polygonal number theorem of Pierre de Fermat. The Jacobi thetafunctions, frequently applied in the study of hypergeometric series, were named in his honor. In developments of the theory of elliptic functions, modern authors mostly follow Karl Weier- strass. The notations of Weierstrass’s elliptic functions based on his p-function are conve-nient, and any elliptic function can be expressed in terms of these. The elliptic functionsintroduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are morecomplex but important both for the history and for general theory. 5 Theory 1. Elliptic Integrals There are three basic forms of Legendre elliptic integrals that will be examined here; first, second and third kind. In their most general form, elliptic integrals are presented in a formreferred to as incomplete integrals where the bounds of the integral representation rangefrom0≤sinφ≤1or0≤φ≤π/2. a) First Kind: Theincomplete elliptic integral can be written as F(sin φ, k)=/integraldisplay sinφ 0dt/radicalbig (1−t2)(1−k2t2),0≤k2≤1 (3.1) 0≤sinφ≤1 byletting t=sinθ,Eq. 3.1 becomes F(φ, k)=/integraldisplayφ 0dθ/radicalBig (1−k2sin2θ),0≤k2<1 (3.2) 0≤φ<π 2 The parameter kis called the modulus of the elliptic integral and φis the amplitude angle. Thecomplete elliptic integral is obtained by setting the amplitude φ=π/2or sinφ=1,the maximum range on the upper bound of integration for the elliptic integral. F/parenleftbigg φ=π 2,k/parenrightbigg =F(sinφ=1,k)=K ( k)=K (3.3) Acomplementary form of the elliptical integral can be obtained by letting the modulus be (k/prime)2=1−k2(3.4) 6 If we let v=tanθand in turn dv=sec2θd θ=( 1+ v2)dθ,then F(φ, k)=/integraldisplaytanφ 0dv (1 +v2)/radicalBigg/parenleftbigg 1−k2/parenleftbiggv2 1+v2/parenrightbigg/parenrightbigg =/integraldisplaytanφ 0dv√1+v2/radicalbig (1 +v2−k2v2 =/integraldisplaytanφ 0dv/radicalbig (1 +v2)(1 + k/primev2)(3.5) The complementary, complete elliptic integral can then be written as F/parenleftbigg φ=π 2,k/prime/parenrightbigg =F(sin φ=1,k/prime)=K ( k/prime)=K/prime(3.6) b) Second Kind: E(φ, k)=/integraldisplaysinφ 0/radicalBigg 1−k2t2 1−t2dt, 0≤k2≤1 (3.7) or its equivalent E(φ, k)=/integraldisplayφ 0/radicalbig 1−k2sin2θd θ , 0≤k2≤1 (3.8) 0≤φ≤π 2 And similarly, the complete elliptic integral of the second kind can be written as E/parenleftbigg φ=π 2,k/parenrightbigg =E(sin φ=1,k)=E ( k)=E (3.9) and the complementary complete integral of the second kind E/parenleftbigg φ=π 2,k/prime/parenrightbigg =E(sin φ=1,k/prime)=E ( k/prime)=E/prime(3.10) 7 c) Third Kind: Π(φ, n, k )=/integraldisplaysinφ 0dt (1 +nt)2/radicalbig (1−t2)(1−k2t2), 0≤k2≤1 (3.11) or its equivalent Π(φ, n, k )=/integraldisplayφ 0dθ (1 +nsin2θ)/radicalBig (1−k2sin2θ),0≤k2≤1 (3.12) 0≤φ≤π 2 8 Computer Algebra Systems More than any other special function, you need to be very careful about the arguments yougive to elliptic integrals and elliptic functions. There are several conventions in common use in competing Computer Algebra Systems and it is important that youcheck the documentation provided with the CAS to determine which convection is incorporated. Function Mathematica Elliptic Integral of the first kind, F[φ|m] EllipticF[ φ,m] -amplitude φand modulus m=k2 Complete Elliptic Integral EllipticK[ m] of the first kind, K(m) Elliptic Integral of the second kind, E[φ|m] EllipticE[ φ,m] -amplitude φand modulus m=k2 Complete Elliptic Integral EllipticE[ m] of the second kind, E(m) Elliptic Integral of the third kind, Π[n;φ|k] EllipticPi[ n, φ,m] -amplitude φand modulus m=k2 Complete Elliptic Integral EllipticPi[ n,m] of the third kind, Π(n|m) Potential Applications Determining the arc length of a circle is easily achieved using trigonometric functions, however elliptic integrals must be used to find the arc length of an ellipse. Tracing the arc of a pendulum can be achieved for small angles using trigonometric functions but to determine the full path of the pendulum elliptic integrals must beused. 9 Relations and Selected Values of Elliptic Integrals Complete Elliptic Integrals of the First and Second Kind, K, K/prime,E,E/prime The four elliptic integrals K,K/prime,E,andE/prime,satisfy the following identity attributed to Legendre KE/prime+K/primeE−KK/prime=π 2(3.13) The elliptic integrals KandEas functions of the modulus kare connected by means of the following equations: dE dk=1 k(E−K) (3.14) dK dk=1 k(k/prime)2[E−(k/prime)2K] (3.15) Incomplete Elliptic Integrals of the First and Second Kind, F(φ, k),E(φ, k) It is convenient to introduce another frequently encountered elliptic integral which is related to EandF. D(φ, k)=/integraldisplayφ 0sin2θ ∆dθ=F−E k2(3.16) where ∆=/radicalbig 1−k2sin2θ (3.17) Therefore F=E+ k2D (3.18) 10 Other incomplete integrals expressed by D,E,andFare /integraldisplayφ 0cos2θ ∆dθ=F −D (3.19) /integraldisplayφ 0tan2θ ∆dθ=∆tanφ−E (k/prime)2(3.20) /integraldisplayφ 0dθ ∆cos2θ=∆tanφ+k2(D−F) (k/prime)2(3.21) /integraldisplayφ 0sin2θ ∆2dθ=F−D (k/prime)2−sinφcosφ (k/prime)2∆(3.22) /integraldisplayφ 0cos2θ ∆2dθ=D +sinφcosφ ∆(3.23) /integraldisplayφ 0∆tan2θd θ =∆ tanφ+F−2E (3.24) Special Values of the Elliptic Integrals E(0,k)=0 (3.25) F(0,k)=0 (3.26) π(0,α2,k)=0 ( −α2=n) (3.27) E(φ, k)= φ (3.28) F(φ, k)= φ (3.29) π(φ, α2,0) = φ (ifn=0 ) (3.30) =arctan/parenleftBig/radicalbig (1−α2)tanφ/parenrightBig √1−α2, ifα2<1 (3.31) =arctanh/parenleftBig/radicalbig (α2−1) tan φ/parenrightBig √α2−1, ifα2>1 (3.32) 11 K(0) = K/prime(1) = π/2 (3.33) E(0) = E/prime(1) = π/2 (3.34) E(φ,1) = sin φ (3.35) F(φ,1) = ln(tan φ+secφ) (3.36) π(φ, α2,1) =1 1−α2/bracketleftBigg ln(tan φ+secφ)−αln/radicalBigg 1+αsinφ 1−αsinφ/bracketrightBigg (3.37) ifα2>0,α2/negationslash=1 =1 1−α2[ln(tan φ+secφ)+|α|arctan( |α|sinφ)] ifα2<0 Differentiation and Integration with Respect to the Modulus k ∂F ∂k=k (k/prime)2/bracketleftbigg F−D−sinφcosφ ∆/bracketrightbigg (3.38) ∂E ∂k=−kD (3.39) ∂D ∂k=1 k(k/prime)2/bracketleftbigg F(φ, k)−D(φ, k)−sinφcosφ ∆−D(φ, k) k/bracketrightbigg (3.40) /integraldisplay Fkd k =E ( φ, k)−(k/prime)F(φ, k)−(1−∆)cotan φ (3.41) /integraldisplay Dkd k =−E(φ, k) (3.42) /integraldisplay Ekd k =1 3(1 +k2)E(φ, k)−(k/prime)2F(φ, k)−(1−∆)cotan φ(3.43) 12 Jacobi’s Nome qand Complementary Nome q1 The nome qand the complementary nome q1are defined by q=q(k)=exp[−πK/prime/K] (3.44) and q1=q(k/prime)=exp[−πK/K/prime] (3.45) Therefore lnqlnq1=π2(3.46) Tocompute the nome qwesetk=sinαand introduce the parameter /epsilon1defined by 2/epsilon1=1−√cosα 1+√cosα(3.47) bymeans of which we have q=/epsilon1+2/epsilon15+1 5/epsilon19+150/epsilon113+1707/epsilon117(3.48) The above series converges rapidly provided 0≤α≤π/4or0≤k≤1/√ 2.F o r π/4≤α<π / 2or1/√ 2≤k≤1,let 2/epsilon11=1−√ sinα 1+√ sinα(3.49) and compute the complementary nome q1bymeans of q1=/epsilon11+2/epsilon15 1+1 5/epsilon19 1+150/epsilon113 1+1707/epsilon117 1(3.50) 13 2. Elliptic Functions There are several types of elliptic functions including the Weierstrass elliptic functions as well as related theta functions but the most common elliptic functions are theJacobian elliptic functions, based on the inverses of the three types of elliptic integrals. 1.Jacobi elliptic functions: The three standard forms of Jacobi elliptic integrals are denoted as sn(u, k),c n(u, k)anddn(u, k)and are the sine, cosine and delta amplitude elliptic functions, respectively. These functions are obtained byinverting the elliptic integral of the first kind where u=F(φ, k)=/integraldisplay φ 0dθ/radicalbig 1−k2sin2θ(3.51) where 0<k2<1andkis referred to as the elliptic modulus of uandφ,the upper bound on the elliptic integral is referred to as the Jacobi amplitude ( amp). The inversion of the elliptic integral gives φ=F−1(u, k)=amp(u, k) (3.52) and from this we can write sinφ=sin(amp(u, k)) =sn(u, k) (3.53) cosφ=cos(amp(u, k)) =cn(u, k) (3.54) /radicalBig 1−k2sin2φ=/radicalBig 1−k2sin2(amp(u, k)) =dn(u, k) (3.55) These functions are doubly periodic generalizations of the trigonometric functions satisfying sn(u,0) = sin u (3.56) cn(u,0) = cos u (3.57) dn(u,0)=1 (3.58) 14 01234567 u/Minus1/Minus0.500.51sn/LParen1u/RParen1dnusnu cnu0K2 K 3K4K Figure 3.1: Plot of Jacobian Elliptic Functions sn(u),cn(u)anddn(u)fork=1/2 In total there are 12 Jacobian elliptic functions, where the remaining 9 can be related to the 3 we have already seen as follows: cd(u)=cn(u) dn(u)dc(u)=dn(u) cn(u)ns(u)=1 sn(u) sd(u)=sn(u) dn(u)nc(u)=1 cn(u)ds(u)=dn(u) sn(u) nd(u)=1 dn(u)sc(u)=sn(u) cn(u)cs(u)=cn(u) sn(u) 2.Weierstrass elliptic functions: The principal difference between the Jacobi and the Weierstrass elliptic integrals is in the number of poles in each fundamentalcell. While the Jacobi elliptic functions has two simple poles per cell and can beconsiders as a solution to the differential equation d 2x dt2=A+Bx+Cx2+Dx3 the Weierstrass elliptic function has just one double pole and is a solution to d2x dt2=A+Bx+Cx2 15 Wewill focus primarily on the Jacobi elliptic function in this course but you should be aware of the Weierstrass form of the function. 3. Theta Functions Theta functions are the elliptic analogs of the exponential function and are typically written as, θa(u, q)where aranges from 1 to 4 to represent the fours variations of the theta function, uis the argument of the function and qis the Nome, given as q=eiπt=eπK/prime/K(3.59) where t=−iK/prime(k) K(k) 16 Assigned Problems Problem Set for Elliptic Integrals and Functions 1. Determine the perimeter of the ellipse 4x2+9y2=3 6 2. Obtain the expression for the length of arc of the ellipse x2+y2 4=1 between(0,2)and(1/2,√ 3).Note that b>a in this problem. Compute the arc length to three decimal places. 3. Compute to four decimal places the following integrals i)/integraldisplayπ/4 0dt/radicalbigg 1−1 2sin2tii)/integraldisplayπ/3 0/radicalbigg 1−3 4sin2td t iii)/integraldisplayπ/2 0dt/radicalbigg 1−1 15sin2tiv)/integraldisplayπ/2 0/radicalbigg 1−80 81sin2td t 4. By means of the substitution x=2sinθ,show that /integraldisplay2 0dx/radicalbig (4−x2)(9−x2)=1 3K/parenleftbigg2 3/parenrightbigg 5. Prove that /integraldisplayπ/2 0dx√ sinx=/integraldisplayπ/2 0dx√cosx=√ 2K/parenleftbigg1√ 2/parenrightbigg 17 6. Given q=1/2,compute k,KandK/primeto six decimal places. 7. Compute to three decimal places the area of the ellipsoid with semi-axes 3,2,and1. 8. The thermal constriction resistance on an isothermal, elliptical disk (a>b )on an insulated isotropic half-space of thermal conductivity kis R∗=kaR=K /radicalBigg 1−/parenleftbiggb a/parenrightbigg2  Compute R∗to four decimal places for the following values of a/b:1,1.5,2,3,4and 5.Use the arithmetic-geometric mean method of Gauss. 9. Derivatives of the elliptic integrals. Show that i)dE dk=E−K k ii)d2E dk2=−1 kdK dk=−E−(k/prime)2K k2(k/prime)2 10. Integrals of the elliptic integrals. Show that i)/integraldisplay Kdk=πk 2/braceleftBigg 1+∞/summationdisplay n=1[(2n)!]2k2n (2n+1)24n(n!)4/bracerightBigg ii)/integraldisplay kKLdk=E−(k/prime)2K iii)/integraldisplay kEdk=1 3[(1 + k2)E−(k/prime)2K] 18 11. Show that i)/integraldisplay1 0kEdk k/prime=/integraldisplay1 0E(u/prime)du=π2 8 ii)/integraldisplay1 0Kdk 1+k=π2 8 12. Derivatives of elliptic integrals with respect to the argument i)d dφF(φ, k)=1/radicalbig 1−k2sin2φ ii)d dφE(φ, k)=/radicalBig 1−k2sin2φ 19 References 1.Abramowitz, M. and Stegun, I.A. ,(Eds.), “Gamma (Factorial) Function” and “Incomplete Gamma Function.” §6.1 and §6.5 in Handbook of Mathematical Functions and Formulas, Graphs and Mathematical Tables, 9th printing, Dover, New York, 1972, pp. 255-258 and pp 260-263. 2.Anderson, D., Vamanamurthy, K. and Vuorinen, M. ,“Functional Inequalities for Complete Elliptic Integrals,” SIAM J. Math. Anal., 21, 1990, pp. 536-549. 3.Anderson, D., Vamanamurthy, K. and Vuorinen, M. ,“Functional inequalities for hypergeometric functions and complete elliptic integrals,” SIAM J. Math. Anal.,23, 1992, pp. 512-524. 4.Arfken, G. ,Mathematical Methods for Physicists, 3rd ed., Section 5.8, “Elliptic Integrals,” Orlando, FL, Academic Press, pp.321-327, 1985. 5.Borwein, J.M. and Borwein, P.B. 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