@Alfred_F If a spherical bullet impacts a plane, the "impression" is a circle (actually a disk) only if the (linear) trajectory is orthogonal to that plane. So, what is the significance of r1, r2, r3?

P.S. It seems that unlike your other problems, this one if very vague.

@Preben Alsholm But in my post,  f(complex) is a "function of a complex number" after evaluation.
Othewise, the simple example of a constant function
f := x -> a;

can be used.

For your example:

'f'(2+3*I).b - b.'f'(2+3*I);

returns <0,0;0,0>.

@Preben Alsholm Actually A, B commute (.) if at least one of them is a complex number i.e. C is in the center of (Expressions, .).
The strange fact is that any function of a complex number commutes (.) with any expression:
f(2+3*I).b - b.f(2+3*I) = 0. 

This reminds me of the following fact:

If the matrix M is defined by

m[i,j] = ( d(1,i) + d(1,j) - d(i,j) ) / 2 

then the set {P_i} can be embedded in R^k  iff  rank(M) <= k.
 

@Andiguys Your C_1 is
{0 <= 0.000023339978559*p1 - 0.85357635872}

Probably you want to replace the previous plot2 with:

plot2 := plot([solve(rhs(C_1[])), t, t = -0.2e9 .. 0.1e9])

I don't understand why you complicate things for such a simple problem.

@Andiguys 

plot1:=plot(TRC(p1), p1 = 0 .. 60000):
plot2:=plot(rhs(C_1[]), p1=10000..60000, thickness=3,color=blue):
plots:-display(plot1,plot2);

Not very useful because C_1 has a small slope.

But you don't need it as I said.

@Andiguys You have a simple concave function TRC in one variable (p1);  see  simplify(TRC(p1)).
You can compute the exact max using

maximize(TRC(p1), location);

                  33379441.96, {[{p1 = 24171.82839}, 33379441.96]}

If you want the max value in another interval [a,b], it will be   33379441.96 if  24171.82839 is in [a,b]
and max(TRC(a), TRC(b)) otherwise.

@Carl Love  Unfortunately, Maple is not able to compute inf, liminf, sup,limsup [Mathematica does!]. When the limit of a bounded function does not exist, limit could return a range a..b including the the image of the function.

For example,

limit(sin(n)+cos(n), n=infinity);

returns -2..2  but liminf and limsup are -sqrt(2), sqrt(2)  (even for sequences).

@Preben Alsholm Yes, as it should, but the fact that assuming makes it fail must be investigated.

@acer  simplify fails if  assuming n::posint   is appended!!!

@Carl Love The simplified result is (6*n - 5) / 16.
It is frustrating that Maple needs a lot of help to obtain it.

@Preben Alsholm  The maplesoft download page (www.maplesoft.com/support/downloads/) contains the update.

Does the interrupt button (bottom left) work in 2025.1? Try:

a:=nextprime(10^83):
b:=nextprime(10^95):
x:=a*b:
ifactor(x);

@mmcdara  I had the impression that you are interested in the sum of (random) floats.

For the symbolic case, see: evalf random results - serious bug - MaplePrimes

@C_R   The main problem is that if a() is defined from the beginning, Maple gives the wrong solution (Carl's code is a workaround).

restart;
a := z-> 1+2*Heaviside(z):
de:= Diff(a(x)*Diff(u(x),x),x) = -1:
bc:= u(-1)=0, u(1)=0: 
sol:= dsolve({de,bc}, u(x));