Here, about 10 h ago, Cesaero R presented a super 28-sd y(2) as a zero of #2^y=2+1/y#. How do you find super strings y(b) as zeros of #b^y=b+1/y, b = 10, e and pi#? How do you prove uniqueness or otherwise of y(b)?
Making separate graphs for
approximation(s) to zero(s) of the given equation as y at the
The second equation xy=1 represents two branches of a rectangular
hyperbola (RH), fanning to infinity, in Q1 and Q3, in between axes as
When b > 1, the first (exponential) graph cuts the RH in Q1, passes
through ((x, y) = (0, 1) and (-(b-1), 0), to meet the RH again in Q3. ,
x-axis is asymptotic to RH and, below x-axis, y = - b is
asymptotic to the exponential graph. So, there are two zeros with
Here, the specified b values are all > 1.
With the limited facilities in my processor,, I could make only 10-sd
approximations, for the positive roots, in each case. I have used
the following iterative formula ( using logarithms ).
My approximations are:
The errors in
The interested reader could hunt for 28-sd approximations, using
The negative zeros could emerge from yet another iterative
The suggested iteration formula that gave the given answers for the
negative roots is
The use of log anywhere here might not give the second zero..
Another iterative form has to be tried.