Yum... indeed, I have Mathematica 5.
I have a question about [2.02].
PF(n, x) = e^(HurwitzZeta'(-n, x)-Zeta'(-n))
= e^(HurwitzZeta'(-n, x)) / e^(Zeta'(-n))
When we refer to the derivative of the HurwitzZeta function and the regular Zeta function, we're referring to the derivative with respect to n, correct? If so then I will make a numerical computational derivative of PF(n, x) to get an idea of what it might look like.
d/dn Zeta(-n) ~ (Zeta(-(n+Dn))-Zeta(-(n-Dn)))/(2Dn)
d/dn HurwitzZeta(-n, x) ~ (HurwitzZeta(-(n+Dn), x)-HurwitzZeta(-(n-Dn), x))/(2Dn)
And then work from there...
Yeah, the derivative is on n; I figured that there is no ambiguity in writing
z' because the derivative over x is basically another (normal) zeta function. The problem with doing it with Mathematica's built in zeta function is that the function is loaded with bugs (at least in 4.2, which is what I have; might have been fixed in 5), and with derivatives for non-real arguments, it's also slow. Instead, here is what I have done so far:
http://www.3dap.com/hlp/hosted/procyon/misc/PowerFactorial.mThis is a really messy WIP but it mostly works, although some things are definitely not being done in the most efficient way possible. The package can be loaded using << (Get) and contains the functions PowerFactorial[n,x] and LogPowerFactorial[n,x] (the second of these is analogous to LogGamma). For example, you can try Plot[Abs[PowerFactorial[1,x]],{x,-2,2}] or Plot3D[Re[PowerFactorial[3/2,x+I y]],{x,-2,2},{y,-2,2}] to see how it works.
The fastest method for numerical calculation is definitely the [7.01]/[7.02] asymptotic power series; an arbitrary accuracy goal cannot be specified for this but it still more than good enough for getting graphs of the function. [5.03] (the plana expansion integral) also seems to be fairly fast and can be made accurate to any required precision. These are the two methods I implemented into that package. I have not been able to get the function options to work correctly, so you will need to manually edit the default method in the package file to switch between the methods (look for the Options definition part and change it between PlanaIntegral and AsymptoticSeries).
The [2.03] difference equation needs to be used for both of these methods since the formulas are convergent/accurate only in parts of the x plane. The two formulas that are valid for all x, the [4.03] quasi-WFT product and the [7.03] limit, converge way too slowly to be of any use numerically, so this seems to be the only way to do it.
CP, have you done any engineering math? Convolution, for example?
You mean the convolution integral transform? I frequently run into integrals of that type but have only encountered the general transform once or twice. I've done all my stuff from a pure math, classical analysis-based perspective but am familiar with some physics and number theory applications.
