I have only seen big O show up in series solutions, as in 

series(sin(x),x)

I've never seen it before show up in result of solve

restart;
eq:=x=p*(a*ln(p+sqrt(p^2-2))+2*_C1)/(2*sqrt(p^2-2));
sol:=solve(eq,p);

What does it actually mean when the solution has  O(RootOf(....))?  

Should not result of solve be exact? isn't having big O means an approximation?

Maple 2022.2 on windows 10

Could you suggest a more elegent way to remove any entry in piecewise which has undefined in it?

expr:=(s+exp(-Pi*s)-exp(-2*Pi*s))/(s*(s^2+2*s+2));
inttrans:-invlaplace(expr,s,t);
Y:=convert(%,piecewise);
#remove all entries in piecwise which has undefined

To obtain this

I can;t just apply select on piecewise. So currently I convert piecwise to list of lists, each sublist has the 2 entries you see above in each row.

Next apply select. Then use piecewise again on the result. This works, but wondering if there is a better way.

Attached worksheet.

Download remove_undefined.mw

Another very strange result from simplify.  Consider

simplify(a*f(x) + b*f(x) + a*g(x))

Which is what is expected. Now 

expr := a*f(x) + b*f(x) + a*g(x) + 1/(a*f(x) + b*f(x) + a*g(x));
simplify(expr)

gives

why?  I expected (a+b)*f(x) also in the first expression. Compare for reference the same thing in Mathematica where it does them both the same way:

You see it simplified it to (a+b)*f(x)+a*g(x) in both places as expected.

I tried using size option for simplify, but it had no effect.

How can one obtain same result shown above using Maple? And why it does not automatically produce this result?

Maple 2022.2

I give up.

Why 

restart;
the_rule:=A::anything+B::anything=A*B;
applyrule(the_rule,a+b);

returns 0?

I was expecting a*b

No clue from help what I am doing wrong.

Maple 2022.2

trace gives this

restart;
the_rule:=A::anything+B::anything=A*B;
trace(applyrule);
applyrule(the_rule,a+b);


{--> enter applyrule, args = A::anything+B::anything = A*B, a+b
                        answer := a + b

                             i := 

                           i := a + b

                          answer := b

                             i := b

                          answer := 0

                             i := 0

                          answer := 0

I am not sure why it is doing the above still. 

I think I will stick to evalindents and subsindets as I do not understand applyrule very well.

Given an expression, I want to do an operation each time the pattern  f(arg1)+f(arg2) is found by replacing it by f(arg1+arg2). Regadless of how many there are. For example

f(A)+f(B) -> f(A+B) and  f(A)+f(B)+f(C) -> f(A+B+C)  and so on. But here is the catch, there could be anything else in the expressions. These will be left unchanged. 

So f(A)+f(B)+x -> f(A+B)+x

I can do it in Maple only when the input is exactly f(A)+f(B)  when the input is f(A)+f(B)+f(C) and so on.

But this is not practical as I need to make new type for each case.

I need a general way that will work for any expression like in the above example.

I am now using evalindets, but I do not know how to tell it the type for the general pattern of  f(n1)+f(n2)+.....+f(nn) to replace these with f(n1+n2+....nn).

For reference, this is code in Mathematica I am trying to translate to maple.

expr = Sin[x] + f[A] + f[B] + 10*Exp[x]/13 + Cos[x] + f[c] + f[10*c];
expr //. f[a_] + f[b_] :> f[a + b]

In the above //. means repeated replacement. So it will keep replacing the same pattern over and over and this works regardless of where f(a)+f(b) show up. They can be anywhere in the sum.

I wish I can the same in Maple using evalindets. I tried patmatch also, and same problem. Which is how to make it general. This is what I tried

expr:=f(A)+f(B);
evalindets(expr,`&+`('specfunc(f)','specfunc(f)'),F->f( op([1,1],F) + op([2,1],F) ) );

Which works

But to detect expr:=f(A)+f(B)+f(C) it would need new code

expr:=f(A)+f(B)+f(C);
evalindets(expr,`&+`('specfunc(f)','specfunc(f)','specfunc(f)'),F->f( op([1,1],F) + op([2,1],F) + op([3,1],F) ) );

Obviously this approach will not work. It will also fail once a new term is added in between. 

But how to extend this to the general case of

expr:=sin(x)+f(A)+f(B)+10*exp(x)/13+cos(x)+f(C)+f(10*C);

Is there a way to tell Maple to apply the pattern over and over like with Mathematica so it works for general case? I need to try to do this using either patmatch or evalindets. Ofcourse I can do it the hard way, by iterating over the expression and collecting all the f() and add their arguments one by one each `+` subtype. But that is now what I looking for.  

There should be something similar to how it is done in Mathematica, but using Maple command. Notice that the Mathematica example will work regadless of where the f(a)+f(b) shows up.

expr = Sin[x] + f[A] + f[B] + 10*Exp[x]/13 + Cos[x] + f[c] + 1/(f[10*c] + x + f[99])
expr //. f[a_] + f[b_] :> f[a + b]

ps. may be I need to use subsindets['nocache'] need to look more into it.