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Is there a compact form for n-tuple relativistic additions of
velocities?
Posted:
May 29, 2010 3:00 PM
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I seek advice on whether a certain very long function can be made
significantly more compact and therefore easier to compute.
I am interested in calculating the value of a function which resembles
both a continued fraction and an iterated function, where the basic
unit of the iteration is the right hand side of the formula for
relativistic
addition of two (collinear) velocities:
(1) W_1 = (u + v) / (1+uv) .
u and v are the summed velocities, given as decimal fractions of the
velocity of light C. Ie: u,v range from 0.0 to 1.0 . Their resultant
velocity is W_n.
The essential idea is to divide C into n <equal> velocities (v) and
then
add them sequentially. My procedure is to take the first sum, W, and
then add to it a third identical velocity v, which gives a second
resultant
velocity
(2) W_2 = (W_1 + v) / (1 + vW_1) .
This is an iteration of the basic form in (1).
Similarly, a fourth identical velocity v is added to W_2, giving
(3) W_3 = (W_2 + v) / (1+ vW_2) ,
and so on and on...
This is to be repeated n times, where n becomes very large and
v commensurately smaller. I wish to know the values of this function
as n becomes extremely large. I especially want to know what the
function converges to as n ---> infinity and C/n ---> zero velocity.
A partial glimpse of this procedure as a single large relation
resembles
an unusual variety of continued fraction. I omit rendering it in ASC
II
form here since it rapidly becomes very confusing and opaque.
Any feedback will be very appreciated.
Thanks,
Gene
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