is the unique log-convex analytic continuation of which maintains the recursive relation is the number of permutations of crystals, As of 2014 MathWorks, the owners of Matlab, have not implemented the unreal function. Nor has the open source clone Octave. It has been implemented in Mathematica which I don’t have.

Aside: Octave is not friendly on Windows, but doesn’t take long to load on Linux. By loading into RAM, Octave allows many Matlab scripts to be incorporated into open source software. Programmers need a cheap and easy way to incorporate advanced maths into their software.

Strangely, Matlab does have the complex-valued Riemann Hence my code uses the reflection formula

This formula is problematic at integers, so for those I use the standard “gamma”. The only other problems are near non-trivial zeros, on the critical strip between 0 & 1. For those, I escape the strip using recursion.

I (Tinos Nitsopoulos) copyright zetaGamma.m under GNU.

One application is to investigate along the critical line, This algorithm is always accurate (the selling point), but slow. For example,

>> zetaGamma([0 3 1/2 1/2+i])
ans =
Inf 2.0000 1.7725 0.3007 - 0.4250i
>> abs(ans(end))
ans =
0.5206
>> sqrt(pi/cosh(pi))
ans =
0.5206

From the reflection formula for

That is, produces hyperbolic functions. Hence the appearance of in the numerical example. is the real part of the analytic continuation of from the critical line. The solution to Laplace’s equation is not unique, given only

Aside: As Feynman points out in his lectures (Vol. II Ch. 7), analytic may solve a physics problem. What problem does solve?