One Special Identity between the complete elliptic integrals of the first and the third kind
aa r X i v : . [ m a t h - ph ] F e b One Special Identity between the complete elliptic integralsof the first and the third kind
Yu JiaInstitute of High Energy Physics, Chinese Academy of SciencesBeijing 100049, China
Abstract
I prove an identity between the first kind and the third kind completeelliptic integrals with the following form:Π (cid:18) (1 + x )(1 − x )(1 − x )(1 + 3 x ) , (1 + x ) (1 − x )(1 − x ) (1 + 3 x ) (cid:19) − x x K (cid:18) (1 + x ) (1 − x )(1 − x ) (1 + 3 x ) (cid:19) = ( , (0 < x < − π
12 ( x − / √ xx . ( x < x > complete elliptic integrals, totally of three kinds. The complete elliptic inte-gral of the third kind, Π, being the most complicated one, can be expressedin terms of the complete elliptic integral of the first kind, K , plus elementaryfunctions and Heuman’s Lambda function or Jacobi’s Zeta function. The lasttwo transcendental functions can in turn be expressible from the incomplete elliptic integrals of the first and second kinds.The aim of this note is to establish a simple relation between the completeelliptic integrals of the first and the third kind, for some specific textureof arguments of course. That is, in this special case, the complete ellipticintegral of the third kind can be transformed to the complete elliptic integralof the first kind, plus elementary function at most, without resort to anyother nonelementary function.There are a variety of conventions adopted in literature in defining theelliptic integrals. In this work I find it convenient to use the same definitionsas taken by [2, 4] for the first, second and third kind of complete ellipticintegrals: K ( m ) ≡ Z dt p (1 − t )(1 − mt ) = π F (cid:18) ,
12 ; 1; m (cid:19) ,E ( m ) ≡ Z dt r − mt − t = π F (cid:18) − ,
12 ; 1; m (cid:19) , (1)Π( n, m ) ≡ Z dt − nt ) p (1 − t )(1 − mt ) = π F (cid:18)
12 ; 1 ,
12 ; 1; n, m (cid:19) , where F and F denote Gaussian hypergeometric function, and Appell hy-pergeometric function of two variables, respectively. In most practical ap-plications, the parameter m and the characteristic n are restricted to beless than 1. However, it is worth emphasizing when these arguments exceed1 or even are off the real axis, these elliptic integrals are still well definedmathematically, though become complex valued in general.2he main result states as follows. For ∀ x ∈ R , there exists an identityΠ (cid:18) (1 + x )(1 − x )(1 − x )(1 + 3 x ) , (1 + x ) (1 − x )(1 − x ) (1 + 3 x ) (cid:19) − x x K (cid:18) (1 + x ) (1 − x )(1 − x ) (1 + 3 x (cid:19) = ( , (0 < x < − π
12 ( x − / √ xx . ( x < x >
1) (2)The K and Π functions with this specific arrangement of the arguments, oddas it may look, are not unfamiliar to particle physicists. In fact these func-tions have been encountered in the analytical expressions for the phase spaceof three equal-mass particles [5, 6, 7]. To be precise, it is worth pointing outthat the coefficient of Π function accidentally vanishes in this example. Notethe 3-body phase space can be obtained from cutting the simplest scalar two-loop sunset diagram. For a general sunset diagram with a nontrivial vertexstructure, or if the power of propagators exceeds than 1, the complete ellip-tic integral of the third kind will inevitably arise in evaluating its imaginarypart. Equation (2) can then be invoked to trade the Π function for the K function, thus considerably simplifying the answer.In the aforementioned physical application, simple kinematics enforces0 < x ≤ , for which both the parameter and characteristic of K and Π arepositive and less than 1. This restriction is not necessary for general purpose,therefore it will be discarded in the following discussion.There exist some formulas which transform one Π function to another Πfunction plus a K function [1, 2]. However, the relation designated in (2),which links one Π function to one K function only, is rather peculiar. To thebest of my knowledge, this relation cannot be derived by any known formula,and it also has never been explicitly stated in any published work. For thisreason, I feel it may be worthwhile to report it here.The strategy of the proof is to construct a differential equation satisfiedby the left side of equation (2), called y ( x ) in shorthand. Employing thewell-known differential properties of complete elliptic integrals [1, 4]: dK ( m ) dm = E − (1 − m ) K m (1 − m ) ,∂ Π( n, m ) ∂m = − E + (1 − m )Π2(1 − m )( n − m ) , (3) ∂ Π( n, m ) ∂n = nE − ( n − m ) K + ( n − m )Π2 n (1 − n )( n − m ) , y satisfies the following first-orderdifferential equation: y ′ = y x + 3 x x ( x − x ) . (4)The magic is that, after the differentiation, the complete elliptic integral ofthe second kind cancels, and the elliptic integrals of the first and the thirdkind conspire, in a rather peculiar way, to cluster into the original form.This is a basic type of linear differential equation, and the correspondingsolution is y ( x ) = C ( x − / √ xx , (5)where C is a constant to be determined. Notice the above solution possessesa pole at x = 0 and two branch points located at x = 1 and x = − , and ingeneral one should not expect C will assume a universal value in the entiredomain of x , R . I shall attempt to fix the value of C region by region.First let us consider the case when x belongs to the open interval I =(0 , y ( x ) = 0 for x = ∈ I . Thisinitial value cannot be satisfied unless if C = 0. Also note the right hand sideof (4) is a continuous function of x in this interval. Therefore, by the existenceand uniqueness theorem for linear equation (for instance, see theorem 2.1 inRef. [8]), the differential equation (4) admits the unique solution y = 0 inthis interval.Next I turn to the solution for x ∈ I = ( −∞ , − ). Examining the leftside of Eq.(2), one readily finds y ( x ) = π for x = − ∈ I . This initialvalue can be satisfied only if C = − π . Thus by the theorem of existence anduniqueness, the function in (5) with this value of C constitutes the uniquesolution in the interval I .There still remain two other intervals, I = ( − ,
0) and I = (1 , ∞ )to be investigated. When x resides in I , both K and Π functions becomecomplex-valued, and the left side of (2) turns to be purely imaginary; whereasas x ∈ I , though both K and Π become also complex, the left side of (2)nevertheless is real. There are no simple initial values can be inferred inthese regions. I performed a numerical check with the aid of the computingpackage Mathematica [4], and find the solution (5) with C = − π in thesetwo regions agree with the left side of Eq.(2) to the 14 decimal place.4quation (2) can have some interesting consequences. Here I illustrateone example:Γ( ) √ π = K (cid:18) (cid:19) = 3 − p √ −
92 Π − p √ − , ! (6)= 3 + p √ −
92 Π p √ − , ! − π vuut √ s √ . The analytic expression for K ( ) in term of Γ function is known [3]. However,it is amusing that these strange looking Π functions can also be put in closedform.Lastly, one natural question may be raised– how about the validity ofequation (2) when the domain of x is extended from real to complex? It isa curious question owning to the rich analytic structure of elliptic integrals.Through a numerical study using Mathematica , I find this relation stillholds in most regions of complex plane. It is most lucid to demonstrate thisexamination in plots. The left side of equation (2) still vanishes in an ovalregion internally tangent to a rectangle (0 < Re z < | Im z | < . < Re z < .
74 and | Im z | < . Acknowledgment
I thank Kexin Cao for encouraging me to write down this note. This researchis supported in part by National Natural Science Foundation of China underGrant No. 10605031. 5 eferences [1] P. F. Byrd and M. D. Friedman,
Handbook of Elliptic Integrals for En-gineers and Scientists , Springer Verlag, Berlin (1971).[2] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions ,Dover Publications, New York (1972).[3] I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series, and Prod-ucts , Academic Press, San Diego (2000).[4] Wolfram Research, Inc.,
Mathematica Edition: Version 5.0 , WolframResearch, Inc., Champaign, IL (2003).[5] B. Almgren, Arkiv f¨or Physik , 161 (1968).[6] S. Bauberger, F. A. Berends, M. Bohm and M. Buza, Nucl. Phys. B , 383 (1995) [arXiv:hep-ph/9409388].[7] A. I. Davydychev and R. Delbourgo, J. Phys. A , 4871 (2004)[arXiv:hep-th/0311075].[8] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations andBoundary Value Problems (2nd Edition), John Wiley & Sons , New York(1969). 6 @ Pi - + €€€€€€€€€€€€€€€€€€€€ D -0.5 0 0.5 1Re z-0.5 0 0.5 1Re z -0.50 0.5Im z-0.500.5Im @ Pi - + €€€€€€€€€€€€€€€€€€€€ D -0.5 0 0.5 1Re z Figure 1: Profile of the left side of Equation (2) as a complex-valued function,with the real variable x promoted to a complex variable z .7 @ Pi - + €€€€€€€€€€€€€€€€€€€€ z K - f H z LD @ Pi - + z €€€€€€€€€€€€€€€€€€€€ z K - f H z LD Figure 2: Examination of the validity of second portion of Equation (2) witha complex variable z , where f ( z ) ≡ − π
12 ( z − / √ zzzz