Oh I think we misunderstood each other.
If I remember you have an algorithm which computes the real-analytic (Kneser) superfunction of \( e^x \) and has the same outcome as Dimitrii's Cauchy algorithm.
Now, I guess you can generalize that algorithm to compute the superfunction of \( b^x \) for any base b, that has two non-real fixpoints.
And the question then would be if (complex) b approaches \( \sqrt{2} \) from the upper halfplane, what are the values of the superfunction (say applied to 1/2) and does the imaginary part tend to 0?
If I remember you have an algorithm which computes the real-analytic (Kneser) superfunction of \( e^x \) and has the same outcome as Dimitrii's Cauchy algorithm.
Now, I guess you can generalize that algorithm to compute the superfunction of \( b^x \) for any base b, that has two non-real fixpoints.
And the question then would be if (complex) b approaches \( \sqrt{2} \) from the upper halfplane, what are the values of the superfunction (say applied to 1/2) and does the imaginary part tend to 0?