Formulas for the Approximation of the Complete Elliptic Integrals

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1 Iteratioal Mathematical Forum, Vol. 7, 01, o. 55, Formulas for the Approximatio of the Complete Elliptic Itegrals N. Bagis Aristotele Uiversity of Thessaloiki Thessaloiki, Greece Abstract I this article we give evaluatios of the two complete elliptic itegrals K ad E i the form of Ramauja s type-1/ formulas. The result is a formula for Γ(1/4) 3/ with accuracy about 10 digits per term. Keywords: elliptic fuctios; sigular modulus; Ramauja; Legedre fuctios; evaluatios; costats 1 Itroductio It is kow that (see [1],[3]): / dθ K(x) = 1 x si (θ) = ( 1 F 1, 1 ) ;1;x 0 (1) is the complete elliptic itegral of the first kid. The elliptic sigular modulus k r is defied from the equatio K 1 kr = r. () K(k r ) It is kow that if r Q +, the k r is algebraic umber. The complete elliptic itegral of the secod kid is E(x) = ( 1 F 1, 1 ) ;1;x (3) ad related with K(x) from the formula E(k r )= K(k ( ) r) r 3K(k r ) a(r) + K(k r ). (4)

2 70 N. Bagis The fuctio a(r) is called elliptic alpha fuctio (see [4]). For r N we set K[r] =K(k r ). It is kow that K[r] ca be expressed i terms of products of Γ fuctios, algebraic umbers ad powers of ([7],[9],[10]). The best way oe ca obtai that is by usig the fuctio b(p) = Γ (p) ta(p). (5) Γ(p) It is also kow that if N = r, where ad r are positive itegers the where M (r) is algebraic. The followig values for M (r) are kow: K[ r]=m (r)k[r], (6) M (r) = 1+k r (7) 7M3 4 (r) 18M 3 (r) 8(1 k r )M 3(r) 1 = 0 (8) (5M 5 (r) 1) 5 (1 M 5 (r)) = 56kr (1 k r )M 5(r) (9) These formulas are for fidig K[4r], K[9r], K[5r], which the evaluatio of them deped oly o kowig k r ad K[r]. Note also that oly (7) ad (8) ca be used. The reaso is that modular equatios of higher degree are ot solvable i radicals. I the preset paper we give evaluatio formulas of K ad E i ifiite series usig oly the elliptic sigular modulus k r at poits q = e r, where r positive real. Also we give evaluatio of the costat 1 1 b = Γ 1 4 (10) 4 3/ i about 10 digits per term formula. Our methods cosists Legedre fuctios, ad we ot use the elliptic alpha fuctio a(r). For a same type series that covergig to 1/ oe ca see [11]. Prelimiary Notes The Legedre P fuctio is defied by P μ ν (z) = 1 Γ(1 ν) z +1 ν/ ( F 1 1 z μ, μ +1;1 ν; 1 z ) (11)

3 Approximatio of complete elliptic itegrals 71 Set φ(z) = F 1 ( μ, μ +1;1 ν; z) = The derivatig φ we have φ (z) = z ν/ Γ(1 ν)p μ ν (1 z) 1 z 1 z ν/ Γ(1 ν) (1 z)z 1 z [( 1 μ + ν + (1 + μ)z) Pν μ (1 z) + (1 + μ ν)pν 1+μ (1 z)] (1) If we assume that ( μ) (1 + μ) z (α + β) =g (13) (1 ν)! the βφ(z)+αzφ (z) =g From (11),(1) ad (13) we have the Lemma.1 If α = ( 1+z) 1 μ + ν +z +μz (14) ( μ) (1 + μ) (1 ν)! ν/ z (α +1)= ( 1 μ + ν) z 1 z Γ(1 ν)p 1+μ ν (1 z) 1 μ + ν +(μ +1)z (15) Note. It is also kow that if for a give r N the umber of fudametal discrimiats is h( r) = 1, the (see [9]): K(k r )=K r = 1/6 (k r k r) 4 Gr, (16) r where G r is a product of Gamma fuctios. We kow that (see [], duplicatio formula): k 4r = 1 k r 1+k r (17) Hece i view of (6) ad (7) K[16r] = 1+k 4r K[4r] = 1+k 4r 1+k r K[r]

4 7 N. Bagis But or Hece or k 4r = Settig r 4r we get 1 k4r 1 k = 1 r (1 + k r = ) (1 k r ) 1+k r 1+k r or equivaletly the followig useful Lemma. If r>0, the k 4r = k r 1+k r = (18) K[16r] = 1+k r + k r K[r] 4 K[16r] = 1+ k r K[r] (19) K[64r] = 1+ k 4r K[4r] K[64r] = 3 Mai Results ( 1+k r + ) k r K[r] (0) 8 Theorem ) ( 1 (!) (k r ) [ 4(1 k r ) +1 k r ] = K(k r ) =ϑ 3 (q) (1) where k r = 1 k r, q = e r. Proof. It is kow (see [1]), that P ( 1/) 0 (1 z) = F 1 ( 1, 1 ;1;z ) () hece if we set μ = 3/ ad ν = 0 i Lemma.1 we get the result.

5 Approximatio of complete elliptic itegrals 73 The result of the above Theorem is ot trivial sice the ϑ 3 -fuctio ca be evaluated from the idetity ϑ 3 (q) = = q (3) i which for this case q = e r ad the two costats e ad ivolved. Theorem 3. 4E(k r ) = K(k r) ( 1 ) + (!) (k r) [4(1 kr) +1 kr] (4) Proof. The evaluatio of E(k r )/ follows if we use the formula: P 1/ (1 z) = [E(z) K(z)]. (5) The oe ca arrive with the same method as i Lemma.1 to the desired result. 4 The Applicatio Formula Set ow p = / /4 (6) the k 100 = p + p ad k 100 = p 1/4 + p From the duplicatio formula is k 400 = p 1/4 ad k +p 1/4 400 = 7/3 p 1/8 ( +p 1/4 ) ( +p 1/4 ) 3/4 p 1/8 + p +p 1/ (7) k 1600 = +p 1/4 + 3/4 p 1/8 + p k 6400 = w 0 = = 5/8 ( + p ) 1/4 +4p 1/4 + pp 1/16 + 3/4 + pp 1/8 + p 1/4 + p + 5/8 ( + p ) 1/4 +4p 1/4 + pp 1/16 + 3/4 + pp 1/8 + p 1/4 + p

6 74 N. Bagis Also from Lemma. we have K[6400] = 1 1+ p 1/4 p 1/4 1/4 8 + p +7/8 + p But it is kow that K[100] = ( /4 ( ) 1 b 80 4) K[100] hece we get the ext about 10 digits per term formula for 1 b(1/4): Theorem [ ] 1 ( /4 ) p 1/4 p 1/4 1/ p +7/8 + p 3 1 [ ] (!) (w 0 ) (1 w0) w0 +1/ = Γ 1 4 3/ :(a) ACKNOWLEDGEMENTS. I would like to thak Professor M.L. Glasser for his very useful material ad the precious time sped for me. Refereces [1] M.Abramowitz ad I.A.Stegu, Hadbook of Mathematical Fuctios. Dover Publicatios. New York (197) [] B.C.Berdt, Ramauja s Notebooks Part II. Spriger Verlag. New York (1989) [3] B.C.Berdt, Ramauja s Notebooks Part III. Spriger Verlag. New York (1991) [4] J.M. Borwei ad P.B. Borwei, Pi ad the AGM. Joh Wiley ad Sos, Ic. New York, Chichester, Brisbae, Toroto, Sigapore (1987) [5] I.S. Gradshtey ad I.M. Ryzhik, Table of Itegrals, Series ad Products. Academic Press (1980) [6] E.T.Whittaker ad G.N.Watso, A course o Moder Aalysis. Cambridge U.P. UK (197)

7 Approximatio of complete elliptic itegrals 75 [7] I.J.Zucker, The summatio of series of hyperbolic fuctios. SIAM J. Math. Aa (1979) [8] Bruce C. Berdt ad Heg Huat Cha, Eisestei Series ad Approximatios to Pi. Page stored i the Web [9] D. Broadhurst, Solutios by radicals at Sigular Values k N from New Class Ivariats for N 3 mod 8. arxiv: v3(math-phy) (008) [10] Habib Muzaffar ad Keeth S. Williams, Evaluatio of Complete Elliptic Itegrals of the Fist Kid at Sigular Moduli. Taiwaese Joural of Mathematics Vol. 10, No. 6, pp , (006) [11] N.D. Bagis ad M.L. Glasser, Cojectures o the evaluatio of alterative modular bases ad formulas approximatig 1/. Elsevier, Joural of Number Theory Vol.13, Issue 10, pp , (Oct. 01) Received: Jue, 01