Elliptic Alpha Function
المؤلف:
Borwein, J. M. and Borwein
المصدر:
P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.
الجزء والصفحة:
...
22-4-2019
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Elliptic Alpha Function
Elliptic alpha functions relate the complete elliptic integrals of the first 
and second kinds 
at elliptic integral singular values 
according to
where 
is a Jacobi theta function and
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
 |
(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
J. Borwein has written an algorithm which uses lattice basis reduction to provide algebraic values for 
.
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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