@mthkvv The situation is more complicated. Some authors take different branches, such that the result is complex for nu>1, 0<k<1 (see e.g. Oldham K., Myland J., Spanier J. - An Atlas of Functions, Springer, 2009).
Unfortunately, Maple's documentation does not contain the details.

@acer @Carl Love  thank you both. It would be nice to know whether there was a good reason for this construct, or it's just for fun. In the latter case, let's hope that at least option encrypted will not be used.

Edit. For combinat:-numbpart more similar constructs are used:

kkk := [op(7, eval(combinat:-numbpart))]:
showstat(kkk[4][NumberOfPartitions]);

revealing some undocumented options.

@acer  Thank you, this is very interesting. My impression was that the lexical table is used only for internal purposes.
I wonder why a procedure would be placed here.
Your example would be greatly appreciated.

@Carl Love  The procedure combinat:-partition simply calls partition1(n,m) . This is not an export, local or library proc and I was not able to find it. Do you know something about it?

@Carl Love  Yes, you are right. Yesterday it was late and I had not the time to test it.
Here is a version a bit faster.

NumPartStrict1:=proc(n, m)
  option remember;
  local k;
  if m=0 then return `if`(n=0,1,0) fi;
  if m*(m+1)<2*n then return 0 fi;
  if m>=n then return 1 + thisproc(n,n-1) fi;
  add(thisproc(n-m+k-1,m-k),k=1..m)
end:

CodeTools:-Usage(NumPartStrict1(2000,100));
memory used=159.09MiB, alloc change=76.01MiB, cpu time=2.77s, real time=2.76s, gc time=93.75ms

                  344032826568833636909684517

Please provide a more detailed statement of the problem.

@Aliocha  It's of course possible to use LineInt, but then you must enter the correct complex components for the vector (otherwise the result is always real). As you see, using int is simpler.

The problem is delicate. In Maple there is no "complex diff". The function abs is nowhere complex-differentiable, not even at real points. The abs(1,z) is a compromise between real and complex differentiation. When using diff with abs in a nonreal context, there will be always problems.

@mzivari It's just a simple change of variables, taking into account the constraints.

@mzivari  Yes, it is nonnegative; not easy to prove though (some work is needed for this).
f >= 0 (in the domain) iff  fuv >=0  for u,v >=0,  where:

f := (x, y, z) -> 1 - x^4 + ln(x^4) + z^4 - y^4 - y^4*(ln(z^4) - ln(y^4)):
fuv:=f(sqrt(1+u), sqrt(1+u+v), sqrt(1+2*u+v)); # >=0 for u,v>=0

plot3d(fuv, u=0..10, v=0..10); # graphically should be clear

@chrisc The integral could exist in some distributional sense. If you know how this distribution is defined, maybe Maple could be used to retrive the desired result.

This seems to be a "try to guess what I mean" challenge.

@Carl Love  Warning: modp(a,b) = irem(a,b)  only for a>=0. Also, `mod` could be mods.
 

@mclaine The assumptions are important but unfortunately Maple is far from perfect here.
Note first that
simplify(f__1 - f __2, symbolic);
returns 0, but then you will have to decide in what conditions the equality is actually true.

The equality holds when lambda__g * a > 0, even when lambda__g and a  are complex, but Maple is not able to simplify to 0, not even:

combine(simplify(f__1 - f__2)) assuming lambda__g * a>0;
 

@one man  The method skips zeros too. Using the zer function I have provided, 
zer(3614)=106.546511887030462;
zer(3682)=107.544351296277895;
so, 68 zeros are missing!

I have a procedure that can find all of them, the first stage being to isolate the zeros. After that, fsolve can be invoked.