@nm Current versions do not treat *.mws as classical sheet

I saved your sheet *.wm as classical sheet *.mws - then print(e) in Maple 2017 has not problems

I downloaded the file Maple2024.1WindowsX64Upgrade.exe which is ~ 300 MB and thus can not be a complete version which is more than 3 GB

For your purpose I guess you will need 2 installations of Maple 2024 and will upgrade 1 of these

@dharr 

A good working horse is double exponential integration with oscillating factor to integrate g(r)*sin(a*r + b) over the half line 0 <= r. Unfortunately Maple does not have it but I once translated code for it, https://www.mapleprimes.com/questions/128564-Any-Way-To-Reclaim-NAG-d01akc-Allocated.

Also note that Wminus and Wplus differ only by 2*m*x, so those integral need not be computed twice

PS: the originator writes (in his post) that he can not reply ... strange, I would just create a new account

@dharr The poster seems to consider the expressions as functions of x (without saying something about the range for x). With increasing x the task leads to highly oscillating integrals and Maple is weak on that. Already with Digits=15 and x=0.001 (or Digits=10 and x=1.0) that direct approach does not work. Likewise you may try to estimate the magnitude of the integral to see that it does not contribute to the numerical result

Remark and first suggestion:

You may try to provide the definitions of your function and constants using MMA syntax (as text). In most cases it can be turned into Maple expression using "convert(..., FromMma). This would avoid typing it again. But probably nobody will do the typing for you.

may be it hangs in one of the methods

Maple 2017 finds it

@vv yes, you are right

My humble opinion: as soon as a post has an answer or a reply it should not be possible to delete it (except for very special reasons).

If it would be only to keep that board "clean": I understand the mmcdara's frustration, cleaning up is less important.

Preben Alsholm once wrote a short procedure "PolarForm" doing that (I have no reference, may be he is willing to post it himself)

Your system consists of 6 rational functions in your 6 variables (and the 'free' parameter g), they are quotients of polynomials in that variables.

Thus there are a lot of singularities and I would not expect a solver will handle that without additional work.

One way could be to solve for the numerators only and care for the denominators later.

Left aside the question why positive solutions should exist.

Remark aside: if Digits:=5 then MinimalPolynomial returns X - 3 for that task

Besides Wikipedia (may differ on selected language) there are proofs and overviews at https://www.math.cmu.edu/~bwsulliv/basel-problem.pdf or https://arxiv.org/abs/2010.03953 (Apostol's double integral is quite puzzling)

@Thomas Richard 

Essentially you reduced it to Int( sin(alpha)*alpha^(r), alpha).

One step more would use Int( exp(a)*a^(r), a) and Maple solves it in terms of the Gamma function which seems more simple.