Asymptotic Formulas for Generalized Elliptic-type Integrals

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1 Asymptotic Formulas for Generalized Elliptic-type Integrals S.L. KALLA and VU KIM TUAN Department of Mathematics and Computer Science Faculty of Science, Kuwait University P.O. Box 5969, Safat 13060, Kuwait Abstract Epstein-Hubbell 6] elliptic-type integrals occur in radiation field problems. The object of the present paper is to consider a unified form of different elliptic-type integrals, defined and developed recently by several authors. We obtain asymptotic formulas for the generalized elliptic-type integrals. Keywords Elliptic-type Integrals, Hypergeometric Functions, Asymptotic Formulas. 1. INTRODUCTION Elliptic integrals occur in a number of physical problems 1,5,7,17], and frequently in the form of multiple integrals. One of the integrals being performed, it leads to an integrand which itself involves elliptic integrals. Epstein and Hubbell 6] have treated a family of elliptic-type integrals Ω j k = π 0 1 k cos θ j 1/ dθ, 0 k < 1, j = 0, 1,,... 1 Certain problems dealing with the computation of the radiation field off axis from a uniform circular disc radiating according to an arbitrary angular distribution law,10], when treated with a Legendre polynomial expansion method, give rise to integrals of form 1. For j = 0, 1 formula 1 reduces to and Ω 0 k = Ω 1 k = λ k Kλ, λ Eλ, 3 k1 k CIMA, University of Zulia, Maracaibo, Venezuela 1

2 where λ = k 1k, and Kλ and Eλ are the complete elliptic integrals of the first and second kinds, respectively 1]. Elliptic integral 1 has been generalized and studied by several authors, Kalla 11], Kalla, Conde and Hubbell 1], Kalla and Al-Saqabi 14,15,16], Srivastava et al 0,1], and others 3,4,19,]. Kalla et al 1,13] and Glasser and Kalla 9] have developed a systematic study of the following family of elliptic-type integrals R µ k, α, γ = π 0 cos α 1 θ/ sin γ α 1 θ/ 1 k cos θ µ1/ dθ, 0 k < 1, Rγ > Rα > 0, Rµ >. 4 It can be easily verified that R j k, 1, 1 = Ω jk, 5 and in terms of hypergeometric functions R µ k, α, γ = ΓαΓγ α Γγ1 k F α, µ 1 µ1/ ; γ;. 6 1 k These authors have obtained a number of recurrence formulas, asymptotic expansion for k 1, computer algorithms, integrals and other useful properties. Recently, Srivastava and Siddiqi 0] have given a unified presentation of certain families of elliptic-type integrals in the following form π Λ α,β λ,µ ρ; k = cos α 1 θ/ sin β 1 θ/ 1 ρ sin θ/] λ dθ, 0 1 k cos θ µ1/ 0 k < 1, Rα, Rβ > 0; λ, µ C; ρ < 1. 7 By comparing 6 and 7 we have Λ α,γ α 0,µ ρ; k = Λ α,γ α λ,µ 0; k = R µ k, α, γ. 8 Here we consider an another generalization of the elliptic-type integrals π Λ α,β λ,γ,µ ρ, δ; k = cos α 1 θ/ sin β 1 θ/ 1 ρ sin θ/] λ 1 δ cos θ/] γ dθ, 0 1 k cos θ µ1/ 0 k < 1, Rα, Rβ > 0; λ, µ, γ C; 9 either ρ, δ < 1 or ρ, or δ C whenever λ = m or γ = m, m N 0. We observe that Λ α,β λ,γ,µρ, 0; k = Λα,β λ,0,µρ, δ; k = Λα,β λ,µ ρ; k, 10

3 where Λ α,β λ,µ ρ; k is defined by 7, and contains other families of elliptic-type integrals as its special cases. In this paper, first we express our generalized elliptic-type integral 9 in terms of the Lauricella s hypergeometric function of three variables F 3 D and then obtain its asymptotic expansion as k 1. Corresponding special cases for R µ k, α, γ and Ω j k are also considered. From definition 9 we have Λ α,β. ASYMPTOTIC EXPANSION FOR Λ α,β λ,γ,µρ, δ; k λ,γ,µ ρ, δ; k = 1 δ γ 1 k µ 1/ 11 1 ω β 1 1 ω α 1 1 ρω λ 1 δω γ µ 1/ 1 k ω dω. 0 1 δ k Comparing the integral in 11 with the integral representation of the Lauricella s hypergeometric function of three variables F 3 D see 8] we can express our generalized elliptic-type integral 9 in terms of the Lauricella s hypergeometric function F 3 D as follows Λ α,β λ,γ,µ ΓαΓβ ρ, δ; k = Γα β 1 δ γ 1 k µ 1/ 1 β, λ, γ, µ 1 ; α β; ρ; δ 1 δ,. k F 3 D To obtain asymptotic expansion of Λ α,β λ,γ,µ ρ, δ; k as k 1 we express the Lauricella s hypergeometric function F 3 D in 1 as a double series of the Gauss hypergeometric functions F 3 D β, λ, γ, µ 1 ; α β; ρ; δ 1 δ, n β mn λ m γ n δ = k α β mn m!n! ρm 1 δ m, F 1 β m n, µ 1/; α β m n; k. 13 Using the analytic continuation formula ] for the Gauss hypergeometric functions in 13 we get F 3 D β, λ, γ, µ 1 ; α β; ρ; δ 1 δ, Γα βγµ β 1/ 1 k = k ΓαΓµ 1/ β mn λ m γ n ρk m δk 1/ µ β mn m!n! 1 δ m, β n 3

4 F 1 β m n, 1 α; 1/ µ β m n; k Γα βγβ µ / 1 k µ1/ β µ / mn λ m γ n ΓβΓα β µ / m, α β µ / mn m!n! n δ ρ m F 1 µ 1/, µ α β m n 3/; µ β m n 3/; 1 k,14 1 δ if µ β 1/ is not an integer. Therefore, ΓβΓµ β 1/ ρ, δ; k = β k β 1 δ γ 1 k β µ 1/ Γµ 1/ ΓαΓβ µ / Γα β µ / µ 1/ k µ 1 1 δ γ µ α β 3/ n µ 1/ n 1 k n µ β 3/ n n! Λ α,β λ,γ,µ F 3 D β, λ, γ, 1 α; β µ 1/; ρk, δk 1 δ, k F 1 β µ n /, λ, γ; α β µ n /; ρ, δ 1 δ where F 1 is the Appell hypergeometric function of two variables 8]. Formula 15 can be considered as the asymptotic series for Λ α,β λ,γ,µ ρ, δ; k as k 1 if µ β 1/ is not an integer. Let now µ β 1/ be an integer, say µ β 1/ = ±l, l = 0, 1,,.... First, let µ 1/ = β l. Applying formula ] k F 1 β m n, β l; α β m n; k Γα β m n 1 k βmn = Γβ m nγα m n l β l rmnl 1 α m n l rmnl k r ln r m n l!r! ln1 k Ψ1 m n l r Ψ1 r Ψβ m n r Ψα r ] 1 k β l Γα β m n l 1 β l r m n l r! 1 k r, 16 Γβ m n Γα m n l rr! we obtain Λ α,β λ,γ,β l 1/ ρ, δ; k = β k β 1 δ γ 1 k l 1 β l β mnr λ m γ n 1 α r ρk m δk n k r m,n, m n l r!m!n!r! ln ln1 k Ψ1 m n l r Ψ1 r Ψβ m n r,15 4

5 Ψα r] 1 l l β k l β 1 δ γ l 1 1 β l Let now µ 1/ = β l. If m n < l we have F 1 β m n, β l; α β m n; k β m n rl m n 1 α rl m n r l m n!r! 1 α l r l r! 1 β l r r! 1 k r δ F 1 l r, λ, γ; α l r; ρ, δ Γα β m n 1 k βl = ΓαΓβ l k r ln ln1 k Ψ1 r l m n Ψ1 r Ψβ r l Ψα m n r l] Γα β m n 1 k βmn Γβ l l m n 1 β m n r l m n r! k r. 18 Γα rr! If m n l then k F 1 β m n, β l; α β m n; k Γα β m n 1 k βmn Γβ m nγα m n l β l mnr l 1 α l m n mnr l k r ln m n r l!r! ln1 k Ψ1 m n r l Ψ1 r Ψβ m n r Ψα r ] Γα β m n Γβ m n Therefore, 1 k βl mn l 1 Λ α,β β l r m n l r! Γα m n l rr! k r. 19 λ,γ,βl 1/ ρ, δ; k = β l k β l 1 δ γ n 1 α rl m n λ m γ n β l r δ k ρ m r mn<l r l m n!m!n!r! 1 δ ln ln1 k Ψ1 r l m n Ψ1 r Ψβ r l Ψα m n r l] β k β 1 δ γ 1 k l 1 β l mn<l l m n 1 β mnr λ m γ n 1 α r l m n r! m!n!r! 5

6 ρ1 k m δ1 k n 1 k r β k β 1 δ γ k l 1 1 δ β l β mnr λ m γ n 1 α r ρk m δk n k r m n r l!m!n!r! 1 δ ln ln1 k mn l Ψ1 m n r l Ψ1 r Ψβ m n r Ψα r] β l k β l 1 δ γ mn l mn l 1 λ m γ n β l r m n l r! α mn r l!m!n!r! n δ k ρ m r. 0 1 δ 3. ASYMPTOTIC EXPANSION FOR R µ k, α, γ Asymptotic expansion for R µ k, α, γ can be obtained in similar manner. Indeed, using the hypergeometric representation for R µ k, α, γ R µ k, α, γ = ΓαΓγ α Γγ F 1 γ α, µ 1/; γ; 1 k µ 1/ and formula ] to the hypergeometric function in 1 we have, 1 k R µ k, α, γ = Γγ αγα µ γ 1/ α γ k α γ 1 k γ α µ 1/ Γµ 1/ F 1 γ α, 1 α; γ α µ 1/; k ΓαΓγ α µ / µ 1/ k µ 1 Γγ µ / F 1 µ 1/, µ γ 3/; α µ γ 3/; k, if γ α µ 1/ is not an integer. If µ = γ α l /, l = 0, 1,,..., we have R γ αl 1/ k, α, γ = k α γ l γ α nl 1 α nl k γ α l n l!n! ln ln1 k Ψ1 n l Ψ1 n Ψγ α n l Ψα n l ] 1 k γ α l 1 γ α n 1 α n l n! γ α l n! n 1 k n.3 6

7 If µ = γ α l /, l = 1,,..., we obtain R γ α l 1/ k, α, γ = 1 α γ l 1 k γ α γ α n 1 α n k n α l n l!n! ln ln1 k Ψ1 n l Ψ1 n Ψγ α l Ψα n ] l 1 α γl 4. SPECIAL CASES γ α l n l n! α l n n! k n. 4 From the general formulas established in the previous sections for Λ α,β λ,γ,µρ, δ; k and R µ k, α, γ, one can derive the corresponding asymptotic formulas for other types of elliptic-type integrals by choosing suitable parameter. For example, if one set α = 1/, γ = 1, and µ = j in R µ k, α, γ, it reduces to Epstein-Hubbell elliptic-type integral Ω j k, and then we have Ω j k = k j 1/ 1/ jn 1/ jn k n 1/ j n j!n! ln ln1 k Ψ1 n j Ψ1 n Ψ1/ n j Ψ1/ n j ] 1 k 1/ j 1 j n!1/ n 1/ n 1/ j n! 5. ACKNOWLEDGEMENT 1 k n.5 The authors are thankful to the Research Administration of the Kuwait University for supports under research projects SM 135 and SM 13. References 1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 197. ] M.J. Berger and J.C. Lamkin, Sample calculation of Gamma ray penetration into shelters. Contribution of sky shine and roof contamination. J. Res. N.B.S. 60, ] S. Bromberg, Resultados sobre una integral eliptica generalizada. Rev. Tec. Ing. Univ. Zulia 15i, ] P.J. Bushell, On a generalization of Barton s integral and related integrals of complete elliptic integrals. Math. Proc. Camb. Phil. Soc. 101, ] P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, Heidelberg,

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