| San José State University
Department of Economics
& Tornado Alley
This is an investigation of mathematical structures which contain within themselves a substructure
which is identical to the structure itself. These are particularly easy structures to analyze. The term self-similar mathematical constructs is best explained by an illustration. Consider the
infinite geometric series
S = 1 + x + x² + x³ + …
which can be factored to give
S = 1 + x[1 + x + x² + x³ + … ]
which is equivalent to
S = 1 + xS
Thus the infinite geometric series contains a subunit which is identical to itself.
Under the presumption that the series does converge to a number the last equation can be solved for S to give
Now, going back to the original definition of the series and noting that S is a function of x,
This must be the same as the derivative of S(x)=1/(1-x); i.e., S'(x)=1/(1-x)². Thus
The same procedure can be performed upon the above equation to give
The left-hand side (LHS) of the above equation can be expressed as
A repetition of the previous procedure yields
In terms of self-similarity the above expression may be derived from the equation S(x)=1+xS(x). One differentiation gives
A second derivation gives
S"(x) = S'(x) + S'(x) + xS"(x) = 2S'(x) + xS"(x)
S"(x) = 2S'(x)/(1-x)
The general relation is
Consider the continued fraction
F = 1 + _x___________________________________ 1 + _x______________________________ 1 + _x_________________________ 1 + _x____________________ 1 + _x_______________ 1 + _x__________ 1 + _x____ 1 + …
Since this fraction continues on forever the expression under the first x is the same as F so the continued fraction can be written as
This means F must satisfy the equation
For x=1 these solutions reduce to F=1.6180 and -0.6180. There is no problem accepting +1.6180 as a value for F but -0.6180 is a puzzle even though it satisfies the equation F=1+1/F. These values however are values involved in the Golden Ratio. Their being solutions to F=1+1/F is confirmed as 1+1/1.618=1+0.6180=1.6180 and 1+1/(-0.6180)=1-1.6180=-0.6180.
Now consider the value of the continued fraction as a function of x. Differentiation produces
(To be continued.)
An infinite exponentiation is something which is raised to a power which is something raised to a power ad infinitum> Suppose
This equation might seem a puzzlement as to whether it has any solution other than the obvious one of a=1 and G=1. However a little manipulation turns it into a seemiongly trivial problem. The manipulation is to take the G-th root of both sides giving
Now if we want a value of a that gives G as a solution we need only take the G-th root of G and we have the answer. For example, for G=2, a=2½=√2. Thus
Furthermore since (½)² = 1/4
To verify these relations consider an iterative scheme of the form
The results of the first 20 iterations are:
However everything is not as simple as the preceding. For one thing G1/C does not have a single valued inverse.
Thus 41/4=21/2 so the infinite exponentiation of 41/4 does not converge to 4, instead it converges to 2.
Also the infinite exponentiation of the cube root of 3 ; i.e.,
should give 3 as a result but the iteration scheme does not converge to 3, instead it converges to a value of 2.478.... which is a lower value of G such that G1/G is also equal to the cube root of 3.
The function G1/G reaches its maximum for G=e, the base of the natural logarithms 2.71828.. At that value of G, G1/G is equal to 1.44466786100977…, call it ζ. This is the maximum value of a for which infinite exponentiation has a finite value and
For details see Maximum.
Thus the maximum value an infinite exponentiation can converge to is e and the maximum base for the infinite exponentiation is ζ=1.44466786100977 so this is why the procedure worked for G=2 and G=1/2 but not for G=3.
(To be continued.)
The maximum of the function is found by finding the value of G such that the derivative is equal to zero. The derivative is found for a function in which its argument variable appears in more than one place is to get the derivative for the variable in each place it appears treating it as a constant in the other places.
For a(G)=G1/G suppose the function is represented as G11/G2, then
Setting this equal to zero
The maximum value of the function G1/G is then e1/e.
The Indian mathematician S. Ramanujan was interested in structures such as
which he called infinitely nested radicals.
Consider the simple case of such structures
This infinitely nested radical contains a substructure which is identical to the overall structure; i.e.,
H must satisfy the equation
For x=1, H must be such that
which has the solutions
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