(04/23/2015, 11:15 PM)JmsNxn Wrote: I've been trying to follow this thread, and finally I have something to contribute. For the bases it might be easier to use my expansion of this function. See http://arxiv.org/pdf/1503.07555v1.pdf

I came up with a holomorphic expression for for these bases. It's a fast converging expression as well. It is not as messy as a Taylor series expansion of this same function. It's also a single holomorphic expression for all , greatly reducing computational time.

This is for the periodic/pseudoperiodic extension of tetration (regular koenigs iteration) for bases

The expression isn't so easy to write out:

Unfortunately, I cannot make it work.

I made this code in PariGP (I attached the file)

Code:

`\p 64`

\\base

a=1.2

\\Delta w, for integration step.

\\To do: iterate until converging precision.

Dw=.01

\\Value of w where the integration/sum will be truncated

\\To do: use only the necessary number of terms to achieve desired precision

w_mx=14

\\Value of n where the summation (inside the integral) will be truncated

\\To do: use only the necessary number of terms to achieve desired precision

n_mx=100

\\Value of n where the summation (outside the integral) will be truncated

\\To do: use only the necessary number of terms to achieve desired precision

m_mx=100

\\n° w values to be evaluated.

\\depends on the integration step Dw

n_w=truncate(1+(w_mx-1)/Dw)

\\Value of w, as function of vector index

\\Integration from 1≤w≤∞ will be truncated at w_mx

w(i)=1+(i-1)*Dw

\\n° rows

\\used as index for the summation

\\rows=n_mx

\\To do: check if some z cause errors.

Invgamma(z)=1/gamma(z)

\\precalculation of ⁿ⁺¹a

\\xa=˟a=ⁿ⁺¹a

\\xa[1]=°⁺¹a=a

Size_xa=max(n_mx,m_mx)

xa=vector(Size_xa,n,a);

for (n=2,Size_xa, xa[n]=a^xa[n-1]);

\\precalculation of factorial

InvFact_n=vector(Size_xa,n,1/factorial(n-1));

\\Integrating Factor, function of w

\\∫ IntSum.w⁻z dw

IntSum=vector(n_w,i,{

sum(n=1, n_mx,

xa[n]*InvFact_n[n]*(-w(i))^(n-1), 0.)

})

Integral(z)=sum(n=1,n_w-1, (IntSum[n]*w(n)^(-z)+IntSum[n+1]*w(n+1)^(-z))/2*Dw )

\\To do: check division by zero.

\\Tetration ^za=za(z)

za(z)={Invgamma(1-z)

*(sum(m=1,m_mx,

xa[m]*(-1)^(m+1)*InvFact_n[m]/(m-z))

+Integral(z))

}

There is a problem with the integral. for values w>14, it start growing very fast, and cannot be computed.

Anyways, I truncated the summations and the integral limits, before it starts diverging, to see what I get. Here is the same expression with the variables I used in the PariGP code:

These are the names of the functions in the code:

It produces something close to the right answer, but ,

I have the result, but I do not yet know how to get it.