Relativistic Speed Composition Formulas

Someone was asking about how we can be sure the correct relativistic
speed composition formula is (u+v)/(1+uv) rather than, say
 
      sin{ arctan[ tan(arcsin(u)) + tan(arcsin(v)) ] }

I'm not aware of any particular motivation at the present time to 
seek a new addition formula for speeds, at least not on the 
macroscopic scale, because the relativistic rule works splendidly, 
and is the only rule consistent with the overall relativistic 
structure that has been so successful at describing and predicting 
physical phenomena.  On the other hand, it's sometimes interesting 
to review the simple algebraic equations associated with relativity 
and compare them - from a purely formalistic standpoint - with other 
functions of the same general class, to clarify what distinguishes 
the formulae that work from those that don't.

Letting v12, v23, and v13 denote the pairwise velocities (in geometric 
units) between three co-linear particles P1, P2, P3, a composition 
formula relating these speeds can generally be expressed in the form

                 f(v13)  =  f(v12) + f(v23)

where f is some function that transforms speeds into a domain where
they are simply additive.  It's clear that f must be an "odd" 
function, i.e., f(-x) = -f(x), to ensure that the same composition 
formula works for both positive and negative speeds.  This rules out 
transforms such as f(x) = x^2,  f(x) = cos(x),  and all other "even" 
functions.  

The general "odd" function expressed as a power series is a linear
combination of odd powers, i.e.,

       f(x)  =  c1 x  +  c3 x^3  +  c5 x^5  +  c7 x^7  +  ...

so we can express any such function in terms of the coefficients 
[c1,c3,...].  For example, if we take the coefficients [1,0,0,...] 
we have the simple transform f(x) = x, which gives the Galilean 
composition formula 
                      v13  =  v12 + v23                       [1]

For another example, suppose we "weight" each term in inverse 
proportion to the exponent by using the coefficients [1, 1/3, 1/5, 
1/7,...].  This gives the transform

         f(x)  =  x + x^3/3 + x^5/5 + ...   =  atanh(x)

leading to Einstein's relativistic composition formula

              atanh(v13) = atanh(v12) + atanh(v23)            [2a]

From the identity atanh(x) = ln[(1+x)/(1-x)]/2  we also have the 
equivalent multiplicative form 

         / 1 + v13 \       / 1 + v12 \  / 1 + v23 \
        ( --------  )  =  ( --------- )( --------- )          [2b]
         \ 1 - v13 /       \ 1 - v12 /  \ 1 - v23 /

which is arguably the most natural form of the relativistic speed 
composition law.  In fact the velocity parameter p = (1+v)/(1-v)
gives very natural expressions for many other observables as well, 
including 

          relativistic doppler shift  =  sqrt(p)


          spacetime interval between 
          two inertial particles each
          1 unit of proper time past
          their point of intersection  =  p^(1/4) - p^(-1/4)

Incidentally, to give an equilateral triangle in spacetime, this 
last equation shows that two particles must have a mutual speed of 
sqrt(5)/3 = 0.745...

Anyway, pressing on with this (admittedly superficial) approach to 
divining the correct speed composition law on purely formalistic 
grounds, consider the set of (odd) coefficients [1,1,1,...], 
corresponding to the "odd geometric series"

     f(x)    =   x + x^3 + x^5 + x^7 + ...     =     x/(1 - x^2)

If we adopt this transform, our composition formula would be

           v13               v12             v23
        ---------    =    ---------   +   ---------             [3]
        1 - v13^2         1 - v12^2       1 - v23^2

Here we see that each speed is normalized by the square of the 
relativistic "gamma" factor.  A variation on this would be to correct
each speed with gamma itself, i.e.,

         v13                   v12               v23
   ---------------   =   --------------  +  ---------------     [4a]
   sqrt[1 - v13^2]       sqrt[1 - v12^2]    sqrt[1 - v23^2]


In trigonometric form this can be written as

      tan(asin(v13))  =  tan(asin(v12)) + tan(asin(v23))        [4b]

and using some basic trig identities it can also be expressed in 
terms of hyperbolic functions as

   sinh(atanh(v13))  =  sinh(atanh(v12)) + sinh(atanh(v23))     [4c]

Notice that [4c] is related to the relativistic formula [2a] simply 
by applying the hyperbolic sine to each term.  The power series 
coefficients of sinh(atanh(x)) are [1, 1/2, 3/8, 5/16,...], which 
seem somewhat less "natural" than any of the previous coefficient 
sets.  Furthermore, both [3] and [4] suffer from the fact that 
although they are singular at arguments of 1, the slope of v13 
does not go to zero as v12 and v23 approach 1.  

To remedy this we could instead apply the INVERSE hyperbolic sine 
to the terms of [2], giving the composition formula

 asinh(atanh(v13))  =  asinh(atanh(v12)) + asinh(atanh(v23))    [5]

which DOES have a zero slope at arguments of 1.  Not surprisingly, 
the discrepancy between [5] and the relativistic formula [2] is about 
the same as the discrepancy between [4] and [2], but in the opposite 
direction.  As a result, the average of the compositions based on 
[4] and [5] is nearly indistinguishable from the relativistic 
composition [2].