## Iterated Logarithmic Functions

Given any two binary operations a(x,y) and b(x,y), suppose there is a
function f such that
f(a(x,y)) = b(f(x),f(y))
A simple example is a(x,y)=x+y, b(x,y)=xy, f(x)=exp(x). Anyway, given
these two operations and the function f, define the auxiliary functions
u(x) = a(x,f(x)) v(x) = b(x,f(x))
It's easy to see that u and v are "conjugates", in the sense that
f(u(x)) = v(f(x))
Slightly less obvious is the fact that the nth compositions of u(x)
and v(x), denoted by u_n(x) and v_n(x) respectively, are also
conjugates, i.e.,
f(u_n(x)) = v_n(f(x))
This identity is sometimes useful in simplifying computations. To
illustrate, consider the case a(x,y)=xy, b(x,y)=x+y, f(x)=ln(x). Here
we have u(x) = x ln(x) and v(x) = x + ln(x). Taking an initial x
value of 10, the first few iterations of u are
10.000000 23.025851 72.223287 309.098522
Beginning with an initial value of f(10)=2.302585 the first few
iterations of v are
2.302585 3.136617 4.279762 5.733660
and we see that f(309.098522) = 5.733660.
Another set of functions to which this method can be applied is
x + y
a(x,y) = x + y b(x,y) = ------- f(x) = tan(x)
1 - xy
In general, any function f() with an "addition rule" can be used.
For example, we can use the functions
a(x,y) = x+y
_______ _______
b(x,y) = x /1 - y^2 + y /1 - x^2
f(x) = sin(x)
This can also be applied to number-theoretic functions. For example,
consider the case a(x,y)=xy, b(x,y)=x+y, f(x)=sopf(x), where sopf is
the "sum-of-prime-factors" function. In this case the auxiliary
functions are u(x) = x sopf(x) and v(x) = x + sopf(x). Beginning
with an initial value of 10, the first few iterations of u are
10 70 980 22540 1036840
Beginning with an initial value of f(10)=7, the first few iterations
of v give
7 14 23 46 71
and we see that sopf(1036840) = 71.

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