@Markiyan Hirnyk 
But you have considered alpha=beta (=1) and the integral is divergent in this case.

@_Maxim_ 

But isn't this normal?
The result of 
Digits:=30: evalf[10](expr);
is and should be the same as:
evalf[10](expr): evalf[30](%);

@Markiyan Hirnyk 

Why? It can be computed by hand using the convolution formula.

@_Maxim_ 

It works, but by chance, for Digits=30 the coefficient of the 1/s term is 0.0  instead of  x.y*10^(-...).

evalf[10]( evalf[40](series(f(1.*s), s = 0, 1)) );
        (1.*10^(-40)*I)/s+O(s)

@Ian Jones 

You must have typos; it is not validated numerically.

@Carl Love 

I mean this:

Download digits.mw

@Carl Love 

But in "most" situations evalf[d1](e)  is enough because Digits is automatically increased by evalf if necessary.

1. The third argument of Cubes must be an m-dimensional list. So, use:
Cubes(3, 3, [a, b, c], h);

2. For an enumeration of N^m  see ?vectoint, ?inttovec

@_Maxim_ 

Nice analysis!

Have you used Maple for the convolution, or you did it by hand?

Cubes:=proc(m::posint,n::posint,a::list,h:=1) # for the cube [a, a+~h] in R^m
local k,u,T,C;
T:=combinat:-cartprod([[seq(k/n*h, k=0..n-1)] $ m ]):
C:=('T[nextvalue]()')$(n^m):
seq( [a+u, a+u+~h/n], u=[C])
end:

Cubes(2,2,[a,b],h);

This is probably because you like the symmetry.
But  [x=3,y=2], [x=2, y=10] etc, are also correct.

@Christian Wolinski 

@arjangash If you really have no problem about optimization, it should be obvious.
Here is a simpler example:

restart;
Z:=(a,b,g)->a+b+g:
constr:=(a,b,g) -> (a>=0,b>=0, a^2+b^2-2*a+1<=g):
Zmax:= g -> Optimization:-Maximize(Z(a,b,g), [constr(a,b,g)])[1]:
plot(Zmax, 0.1 .. 2.2);

@acer 

You are right, I simply forgot about having reported it, sorry! Should I delete the question?

In my opinion the problem is not to find workarounds (which are not complicated) but to understand where exactly is Maple's fault.

Maple can be used to compute the limit of the sequence

n -> sqrt(n) * (sin@@n)(1);

Can MMA compute it?