@torabi 

You must give the general form of your function because it would be difficult (time consumming) to consider the period of an arbitrary function.

@Carl Love 

You simply have a typo (x =s).

@Carl Love 

Yes, but the name too long.

@Carl Love 

Thank you. It is very instructive.

@digerdiga 

combine(((1-t)^2)^q) assuming 1-t > 0;
   (1-t)^(2*q)

(even for a complex q)

@Carl Love 

I am not able to provide such an example. A good candidate should be
isolve((y^2-x^3-1)*(x-2*y)*(y-2*x) = 0);
but Maple does not solve y^2-x^3-1 = 0 and however the flag is not set; it should be clear (for Maple and for us too) that not all solutions have been found e.g. x=2,y=3 [actually this is the only one which is missing].
The reason I asked the question is that I don't know why _SolutionsMayBeLost   has disappeared. It would be nice to have it back.

@Carl Love 

I don't think you are right about _SolutionsMayBeLost . solve has no reason to set it because the system is very simple.
And as I have said, the flag was set in earlier versions (and the answer was the same).

@boblas 

I have already shown this:

prod( wdeg([1,1,0,1,1,0,etc], [x,y,z,t,u,v,etc]), plex(x,y,z,t,u,v,etc) );

If you want to test it, please come with a concrete polynomial P (or a list of monomials) and provide also the sorted version of P.

@Carl Love 

If the starting point is changed from x to  `if`(x<2,x/2, ln(x/ln(x)))  then 2 iterations are enough for any x>0 with accuracy 4e-10 for evalhf.  And actually the accuracy is better; for evalf[30] ==> 1e-17.

W(W(`if`(x<2,x/2, ln(x/ln(x))),x),x):

But of course this is not important because an iteration of W is probably not much more expensive than ln(ln(x)).

Edit.  W(W(x/2,x),x)  has accuracy 1e-7 in the interval 0.3 .. 10

@Carl Love 

Very nice, vote up!
So, you took x as starting point for iterations. Of course there are better ones, but it has the advantage that the first iteration does not contain ln.

@brian bovril 

I did not derive it; it's in

D.A. Barry et al.,  Analytical approximations for real values of the Lambert W-function,
Math. Comput. Simul. 53(2000), 95-103.

It has not a great accuracy but it's simple and is valid for x>0.

limit(LambertW(x)-W, x=infinity) = 0.051...

@zarara 

We have just proved it.
Note that Maple is able to obtain the answer without knowing it:
MultiSeries:-asympt(n^2/2 + 2*n - 1, n, 0);
        O(n^2)

@Carl Love 

You have invested a lot of work and have transformed the problem into a larger project. Congratulations!
I have some remarks/questions.

- If you use the compiler, why do you keep the evalhf part? evalhf is slower and works for smaller integers (in Maple 64 bits).
BTW, the compiler does not accept max with >2 arguments, you must change this.
- I think that the compiled part shoud be used whenever p<pmax, without try/catch. I also don't thing that recursion is useful (I have mentioned this in a previous comment).
- For
#p:=nextprime(8*10^7); n:=p*2-3;  k:=floor(3*p/2);  # answer=50000014
p := 80000023;n := 160000043;k := 120000034;

Binomial needs minutes instead of seconds and a lot of gc. Is this because of recursion? Does the compiler work only once?

In maths we must accept the definitions. Unfortunately it may happen that our prefered vesion is changed in another source. In Maple  the CDF is F(x) = P(X<=x);  I would prefer  "<", but that's it.
I stopped reading the worksheet when I saw that the definitions where not included or discussed.

You said that the quantile results are not correct for the discrete uniform distribution. But it conforms to the definition given in the help page (and widely accepted). So, où est le reproche?

@rlopez 

Why challenging? I'd say that the problem is simply incorrectly or/and incompletely formulated.