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علماء الرياضيات | Elliptic Alpha Function |
 1260
 12:57 صباحاً
date: 22-4-2019 |
Author : Borwein, J. M. and Borwein 
Book or Source : P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. 
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Date: 20-8-2018
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Date: 14-8-2019
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Date: 25-7-2019
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Elliptic alpha functions relate the complete elliptic integrals of the first 
and second kinds 
at elliptic integral singular values 
according to
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(1) |
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![pi/(4[K(k_r)]^2)+sqrt(r)-(E(k_r)sqrt(r))/(K(k_r))](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline9.gif) |
(2) |
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(3) |
where 
is a Jacobi theta function and
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(4) |
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(5) |
and 
is the elliptic lambda function. The elliptic alpha function is related to the elliptic delta function by
![alpha(r)=1/2[sqrt(r)-delta(r)].](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/NumberedEquation1.gif) |
(6) |
It satisfies
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(7) |
and has the limit
![lim_(r->infty)[alpha(r)-1/pi] approx 8(sqrt(r)-1/pi)e^(-pisqrt(r))](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/NumberedEquation3.gif) |
(8) |
(Borwein et al. 1989). A few specific values (Borwein and Borwein 1987, p. 172) are
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(10) |
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(11) |
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(12) |
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(13) |
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(14) |
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(15) |
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(16) |
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![1/2[3-3^(3/4)sqrt(2)(sqrt(3)-1)]](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline47.gif) |
(17) |
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(18) |
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(19) |
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(20) |
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(21) |
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(22) |
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(23) |
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(24) |
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![5/2[1-25^(1/4)(7-3sqrt(5))]](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline71.gif) |
(25) |
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![3[1/2(sqrt(3)+1)-2^(1/3)]](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline74.gif) |
(26) |
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![1/2{sqrt(30)-(2+sqrt(5))^2(3+sqrt(10))^2×(-6-5sqrt(2)-3sqrt(5)-2sqrt(10)+sqrt(6)sqrt(57+40sqrt(2)))×[56+38sqrt(2)+sqrt(30)(2+sqrt(5))(3+sqrt(10))]}](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline77.gif) |
(27) |
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![1/2[sqrt(37)-(171-25sqrt(37))sqrt(sqrt(37)-6)]](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline80.gif) |
(28) |
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![1/2[sqrt(46)+(18+13sqrt(2)+sqrt(661+468sqrt(2)))^2×(18+13sqrt(2)-3sqrt(2)sqrt(147+104sqrt(2))+sqrt(661+468sqrt(2)))×(200+14sqrt(2)+26sqrt(23)+18sqrt(46)+sqrt(46)sqrt(661+468sqrt(2)))]](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline83.gif) |
(29) |
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![7/2-sqrt(7[sqrt(2)7^(3/4)(33011+12477sqrt(7))-21(9567+3616sqrt(7))])](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline86.gif) |
(30) |
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![[1/2(sqrt(29)+5)]^6(99sqrt(29)-444)(99sqrt(2)-70-13sqrt(29))](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline89.gif) |
(31) |
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(32) |
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![(8[2(sqrt(8)-1)-(2^(1/4)-1)^4])/((sqrt(sqrt(2)+1)+2^(5/8))^4).](http://mathworld.wolfram.com/images/equations/EllipticAlphaFunction/Inline95.gif) |
(33) |
J. Borwein has written an algorithm which uses lattice basis reduction to provide algebraic values for 
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REFERENCES:
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.
Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201-219, 1989.
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