Cauchy Integral Experiment

[Kouznetsov]

Hello mike3. I looked at your code.
I would mention two the defects, that may be important:

1. The code does not provide any graphical output.
It is difficult to localize the step responsible for the instability you report.

2. The routine CauchyIntegralConst looks suspicious.
I doubt if it treats well the cutlines of the logarithmic function.
I suggest that you plot the result of the evaluation with CauchyIntegralConst
as function of position of the point of evaluation in the complex plane,
the cut lines are supposed to be seen at such a plot;
and let us compare it to the figure 3 from my paper in Mathematics of Computation.

Can this also be used for the imaginary-periodic bases \( 1 < b < e^{1/e} \)?

I doubt if you can generalize the algorithm to other bases before you make it work well for base b=e.
Also, there should be additional problems with the closure of the contour of integration. They are discussed at
http://math.eretrandre.org/tetrationforu...hp?tid=248

Another question is that if I try using the half-circle contour for the endpoints (as opposed to not updating them at all, which is what I do in the above), the thing diverges wildly. Presumably, the values of the integral around the endpoints should approximate L/2 and conj(L)/2 as the Cauchy integral of a constant function around a half-circle whose center is on the parameter/evaluation point ("a" in the integrand \( \frac{f(z)}{z - a} \)) is K/2, where K is the value of the constant function, and tet is assumed approximately constant at large z = ix, x real. Why is that?

That is because you try to debugg your code in "one piece", without to arrange the routines for the internal tests of your intermediate steps and service functions.
Consider to upload the plot of your approximation of tetration along the imaginary axis, just before and just after the "wild divergence" you report;
then, I hope, it will be possible to reveal the cause of such a behavior.

With 209 nodes, A = 10, 32 normalized iterations followed by 6 unnormalized ones (causes a little "drift"/"sag" of the middle peak but seems to improve accuracy better once this is corrected for via looking for the necessary offset/bias by which to shift the function reconstructed from these imag-axis values through the Cauchy formula so it has value 1 at z = 0) gives a real-axis residual of ~0.00002.

Could you plot this residual as function of the imaginary part of the argument?
How does this residual depend on the order of update of the values approximation tetration in your array?

I did not run the Cauchi with the Simpson approximation of integrals.
I used the approcimation with "rectangles" (second order of approximation)
and, after to get the residual of order of 10^(-5), I switched to the Gauss-Legendre (infinite order of approximation).
In my code, I used to update first the only even nodes: 0,2,4,6,..
(The odd points were updated authomatically due to the symmetry);
after to see that it provides the residual at the level of the rounding errors,
I stoped to play with the order of updates;
I did not test many other ways to choose the order of uprates.
However, I expect, that the final result does not depend on the way we approximate the integrals. If the procedure converges, it converges to the holomorphic function, satisfying the equation tor the superexponential.

How does the residual depend on the step of integration?
How does it depend on the parameter A ?