Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))

[Gottfried]

Hmm. Manually I get a satisfying result today.


First recall my matrix-formula:
\( \hspace{24}
V(x)\sim * B_s = V(s^x)\sim \)

with the hypothesis that the eigensystem-decomposition

\( \hspace{24} B_s = W * E * W^{-1} \)

holds for \( \hspace{24} 1/e^e <s < e^{1/e} \)

Assuming a value \( \hspace{24} t \) so that
\( \hspace{24}
s = t^{1/t} \hspace{24} tl=\log(t) \hspace{24} -1<tl<1 \)
I use for the following my analytical result for the construction of B_s from the Eigensystem.
The basic definition is simply

\( \hspace{24} B_s =\phantom{a}_dV(log(s)) * B \)

where
B = matrix(c^r/r!), c=colindex, r=rowindex, c,r>=0
and there is no complex value involved here, if only s is real and s>0.

But for the eigensystem, the needed parameters t and log(t) are conventionally defined only for the above mentioned range of s.
What if s is beyond the above bounds? We don't have then immediately a valid t available. The empirical eigensystems of truncated matrices Bs of such parameters have partly erratic structure and instability even when increasing the dimension, and powers of its diagonal can only be used in a very limited range for the exponents and unknown approximation quality.


Backed by my previous plots I searched manually for the example
\( \hspace{24} s=7 \)
and found one possible t by binary search and approximation as solution
\( \hspace{24} \begin{eqnarray}
t &&=&& 0.108080950260 + 0.243817409271*I \\
tl &&=&& -1.32163414553 + 1.15353940987*I
\end{eqnarray} \)

I can insert these values for the composition of W, E and W^-1 due to my analytical description for their entries.



Let's denote the above eigensystem-decomposition and indicate the used parameters t and tl :

\( \hspace{24} \begin{eqnarray}
B_t &&=&& W_t * E_t * W_t^{-1} \\
&&=&& (\phantom{a}_dV(1/t) * P^{-1}\sim * XI )* \phantom{a}_dV(tl) * (X * P\sim * \phantom{a}_dV(t))
\end{eqnarray} \)

and the basic formula

\( \hspace{24}
V(x)\sim * B_t = V(s^x)\sim \)

is expanded

\( \hspace{24}
V(x)\sim* (\phantom{a}_dV(1/t) * P^{-1}\sim * XI )* \phantom{a}_dV(tl) * (X * P\sim * \phantom{a}_dV(t)) = V(s^x)\sim
\)

It should be omputed by parts, using associativity:

\(
1)\hspace{24} V(x')\sim = (V(x)\sim* \phantom{a}_dV(1/t) * P^{-1}\sim) = V(x/t-1)\sim \\
\\
2) \hspace{24}
M = ( XI * \phantom{a}_dV(tl) * X) * (P\sim * \phantom{a}_dV(t))
\)

I computed the direct sums of the final matrix-product and its expected result

\( \hspace{24}
V(x')\sim * M = V(s^x)\sim = [1, s^1, s^2, s^3 ,....]
\)

giving the following result in the first four columns (dim=64, rounded to six digits)

Code:

[1.000000, 7.000000, 49.000000, 343.000000 + 8.0307505 E-11*I]

which agrees beautifully with the expected result




So the search for solutions t=h(s) where s out of the bounds [1/e^e .. e^(1/e)], as indicated in the previous graphs, seems to be useful.
Next step would either be, to make some more numerical results available, or to prove the appropriateness of defining B_s via a complex eigensystem, using the real values of complex h(x)-parameters.

The graph gives an impression, where the imaginary zeros of h(x) are located; it does not show, that the real values on the traces seem continuous and to cover all values (possibly below a bound) on each of the "circles". The graph shows some repetitions; I assume, there are infinitely many solutions t for each real s and they are correct values for the eigensystem.

[update] The beautiful result is not so beautiful, if we change some of the parameters; then the problem of branches of logarithm occurs.
Using the formula
V(x)~ * Bs = V(s^x) ~
where Bs was created by the eigensystem-approach, then the simplest example is, using s=7, and the above values for t and tl,
V(1)~ * Bs = V(s)~
which worked fine. However, trying
V(0.5)~ * Bs = Y~
gave the negative root -sqrt(s) in Y[1], and generally
[update]
V(1/k)~ * Bs = V(y)~
where y is s^^(1/k)*exp(2*Pi*I/k), so this problem may be cured by rotating the result appropriately.
[/update]

Trying any integer or fractional iterations m
V(1)~ * Bs^m = Y~
then relating y=Y[1] to the expected result was intractable at a first glance.

So there are some more considerations needed...


Gottfried