# 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)?

##### 1 Answer

#### Explanation:

Making separate graphs for

approximation(s) to zero(s) of the given equation as y at the

common point(s).

The second equation xy=1 represents two branches of a rectangular

hyperbola (RH), fanning to infinity, in Q1 and Q3, in between axes as

asymptotes.,

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

opposite signs...

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

128-bit processor.

The negative zeros could emerge from yet another iterative

method.

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.

When