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PhL
Posts:
12
Registered:
12/13/04
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Laplace limit using Laguerre polynomials
Posted:
Aug 10, 2006 3:10 AM
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I've been posting once about the same topic, but I came up with a new
formulation using associated Laguerre polynomials which makes the
result more satisfactory.
The Laplace limit is the value (Sloane's A033259) for which Laplace's
formula for solving Kepler's equation begins diverging. The constant is
defined as the value lambda of x such that
x exp[sqrt(1+x²)]
------------------- = 1
1 + sqrt(1+x²)
Numerically, lambda = 0.66274341934918...
(see http://mathworld.wolfram.com/LaplaceLimit.html).
I've read in litterature that no series development for lambda is
known. What about these ones for lambda _squared_, based on
Lagrange-Burmann expansion technique ?
Inf
---
\ 2 L[n-1,1,4n]
sqrt(1+lambda²) = 1 + > ----------------
/ n exp(2n)
---
n=1
Inf
---
\ 8 L[n-1,2,4n] - 4 L[n-1,1,4n]
lambda² = > -------------------------------
/ n exp(2n)
---
n=1
with L[n,k,x] associated Laguerre polynomial
(http://mathworld.wolfram.com/LaguerrePolynomial.html).
Seems to be working numerically. The first terms are :
lambda² = 4/e^2 - 8/e^4 + 28/e^6 - 368/3e^8 + 1820/3e^10 - 16184/5e^12
+ ...
This is a convergent alternate series (seems to be in n^(-7/4) but I
can't prove it).
Some residual errors by truncating the second sum at n terms :
n=10 terms : 0.4%
n=30 terms : 0.07%
n=50 terms : 0.02%
PhL
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