Let f be a superposition of two functions f1, f2 . i.e. f(x) = g(f1(x),f2(x)) .

If f1 has period T1 and f2 has period T2 then f will have generally as period the lcm of T1 and T2, provided that T1, T2 are commensurable (i.e. T1/T2 is rational).
In your example we must know Omega. E.g. for Omega=1 the function is not periodic.

CommonPeriod:=(T1::Not(0),T2::Not(0)) -> `if`(type(T2/T1,rational), abs(numer(T2/T1)*T1), infinity);

For example, the function    cos(4*x) + sin((3/2)*x)     will have the period
CommonPeriod( 2*Pi / 4,  2*Pi / (3/2));
      4*Pi

A convert and a correct assumption are needed.

simplify(convert(exp(I*t)^q, exp)) assuming t > -Pi, t <= Pi, q > 0, q < 1;

Why don't you use the monomial order lexdeg([x,y,z], [u,v,w]) ; it is used exactly to eliminate x,y,z.

Your monomial order can be obtained:

t2:=plex(u,v,w);  t1:=wdeg([1,1,1,0,0,0],[x,y,z,u,v,w]);

t:=prod(t1, t2);  # your order

sat:=a -> satisfies(u -> (op([0,0,0],u)=D and op([0,1],u)=a)):
difford:= (e,a) -> max(nops~(map2([op],[0,0,..],indets(e,sat(a))))):

# Example
expr:=(diff(a(x, y), x, y))*x*y+(diff(a(x, y), x, x, y))*x+diff(b(x, y), x, x)-sin(z):
expr:=convert(expr,D):
has(expr,a); # depends on a?
                              true
has(expr,c); # depends on c?
                             false
difford(expr,a);
                               3
difford(expr,b);
                               2

For example, the principal branch with ~ 3 digits accuracy, for x>0:

W = 1.45869*ln(1.2*x/(ln(2.4*x/(ln(1+2.4*x))))) - 0.45869*ln(2*x/ln(1+2*x))

You probably know that  f(n) = O(g(n))  means  limsup( f(n)/g(n), n=infinity) < infinity

provided that f,g>0  and  n -->oo.

Now, Maple does not have limsup; we shall use limit instead (in most cases it's OK).
So, your "equation" is equivalent to

limit( (n^2/2 + 2*n - 1)/n^2, n=infinity ) < infinity;
        1/2 < oo
It is true. What do you want to solve here? Do you mean prove?

LinearAlgebra works fine. You must know that it uses Matrix and Vector (capital M and V) instead of the old matrix and vector.
I suspect that you have used matrix. If this is not the case, please post your example.

The equation is equivalent to

x ln(x) = ln(4) + 2n Pi I <==>  ln(x) exp(ln(x)) = ln(4) + 2n Pi I <==>

ln(x) = W(m, ln(4) + 2n Pi I)   and  Im( W(m, ln(4) + 2n Pi I) )  in (-Pi , Pi].

Here m,n are integers.

But  Im( W(m, ln(4) + 2n Pi I) )  in (-Pi , Pi]   iff  m=0  (for n in Z).

Note that m is an arbitrary integer if all the branches of the logarithm (or equivalently of x^x) are considered.

BIN:=module()
export ModuleApply, Fac, Fac1;
local pmax:=trunc(2^((kernelopts(wordsize)-1)/2));

Fac:=proc(n,m,p)
  local r:=1,k;
  for k from n to m do r := irem(r*k,p) od
end:

Fac1:=proc(n::integer,m::integer,p::integer)::integer;
  option autocompile;
  local r::integer:=1, k::integer;
  for k from n to m do r := irem(r*k,p) od
end:

ModuleApply := proc(N::nonnegint, K::nonnegint, p::prime)  # combined with Carl's
local r:=1, n:=N, k:=min(K, N-K), Bin, fac:=`if`(p<pmax,Fac1,Fac);

Bin:=proc(N,K)  # 0 <=K < p, 0 <= N
local k:=min(K,N-K);
if k<0 then return 0 fi;
fac(N-k+1,N,p)/fac(1,k,p) mod p
end;

if k<0 then return 0 fi;
while k > 0 and r > 0 do 
   r:= irem(r*Bin(irem(n,p,'n'), irem(k,p,'k')), p) 
od;
r
end:

end module:

p:=nextprime(8*10^7): n:=p*2-1:  k:=floor(p/2):
CodeTools:-Usage(BIN(n,k,p));
memory used=2.97MiB, alloc change=0 bytes, cpu time=1.89s, real time=1.88s, gc time=0ns
                            80000022

p:=nextprime(9*10^8): n:=p*2-1:  k:=floor(p/2):
CodeTools:-Usage(BIN(n,k,p));
memory used=2.11KiB, alloc change=0 bytes, cpu time=20.78s, real time=20.78s, gc time=0ns
                           900000010

NOTE. The compiler works for p < 3.037*10^9. For p greater than that, BIN uses the non-compiled Fac but then k should be smaller (or at least the reductions via Lucas' theorem) to have reasonable execution time. The magnitude of N is not important.
 

macro(ff=2*a*b-3*a-2*b-c):
f:=(a,b,c)-> `if`(ff<=30 and ff>=15,ff,0):
A:=Array(0..10,1..5,0..1,f, storage=sparse):
indices(A,pairs); # or simply inspect the Array using the matrix browser

(10, 3, 1) = 23, (7, 4, 1) = 26, (5, 4, 0) = 17, (4, 5, 1) = 17, (9, 3, 0) = 21, (8, 3, 1) = 17, (4, 5, 0) = 18, (6, 4, 0) = 22, (5, 4, 1) = 16, (8, 3, 0) = 18, (10, 3, 0) = 24, (5, 5, 0) = 25, (7, 4, 0) = 27, (5, 5, 1) = 24, (9, 3, 1) = 20, (7, 3, 0) = 15, (6, 4, 1) = 21

Vector needs a procedure (with 1 argument) as initializer.
fenq  is such a procedure but  fsenq(2.1, j)   (actually you mean fseqn) is not; it's an expression.
So, use

psf := Vector(5, j->fseqn(2.1, j));

or maybe the more "sophisticated"

psf := Vector(5, curry(fseqn, 2.1));

The roots of integer numbers are not automatically simplified. Not even 4^(1/2). It is by design.
Unfortunately it is hard to find in the help pages "automatic simplification",  even if the Programming Guide contains information about this important aspect.

plots:-animate(plot3d,
[[y, x, (1/3)*Pi], x = 0 .. 2*Pi*a, y = 0 .. 5*a, coords = spherical, scaling = constrained,axes=none], a = 0 .. 1);

using spherical coordinates.

d1:=3: d2:=3: d3:=5:    x0:=0: y0:=0: z0:=0:
eq:=[x^2+y^2+z^2-d1^2, (x-3)^2+y^2+z^2-d2^2, x^2+y^2+(z-4)^2-d3^2]:
sph:=[x=x0+r*sin(v)*cos(u), y=y0+r*sin(v)*sin(u), z=z0+r*cos(v)]:
Eq:=simplify(eval(eq,sph)):
R:=min(seq(max(solve(e,r)),e=Eq)) assuming real:
plot3d(rhs~(eval(sph,r=R)), u=0..2*Pi, v=0..Pi, color=green);

int( R^3*sin(v)/3, u=0..2*Pi, v=0..Pi, numeric, epsilon=1e-5); #volume
                          23.55847110
# The integral can be computed symbolically but the result is ugly.

Edit: the code is now a bit more general.