@acer 

Yes, but graphically.

For (a^3)^(1/3)

we imagine a pointer pointing at -1. Multiplication by -1 (i.e. adding pi) lets the pointer rotate counter clockwise to 1. Multiplying again by -1 rotates the pointer again counter clockwise to -1 where we startet.
Raising this result to the power of 1/3 rotates the pointer back to pi/3, which points now to a complex number.

For (a^1/3)^3

the pointer rotates first back to pi/3. Rasing this to the power of 3 rotates the pointer forward to -1 where we started.

For a=1 nothing moves in both cases.

@acer 

The context panel addition is interesting, but I will probably not need it very often and it risks to get lost at the next Maple update.

How did this question come up? I wanted to demonstrate why Maple does not simplify (a^3)^(1/3) without assuming positive but it does simplify, or better evaluate, (a^1/3)^3. For this I wanted to use a numeric value on the unit circle and took -1 to quickly convert and manipulate it with Maple. This did not go as swift as I though. All that originated from an ode that was not in standard form. Raising this ode to the power of 1/3 leads to complex derivatives if dy/dx is negative. To bring the ode to the standard form with Maple, simplify/symbolic must be used (vv did it by hand). I am still musing about the mathematical interpretation of this operation (and where such odes in nonstandard form come from).

Anyway, even though I am not in desperate need, the use, look and feel of a context panel entry to complete the set of manipulations (besides magnitude, argument, Re, Im and conjugate) on a complex number would be interesting. How should it be labeled to be consistent with other entries? Polar form, Exponential form or Inert exponential form?

Maybe "convert -> Polar" with two sub menus is sufficient: one producing Maple polar output and another producing inert e output. This could have an educational aspect too. Users would be informed that there is a polar way to do arithmetics and a “PolarForm” output (or the like) for a textbook style output using a new convert extension to be implemented. Users could learn without consulting the help system in the first place and easier understand why Maple is implemented the way it is.

I am not sure how to deal best with complex numbers on the real axis without prior knowledge of this

-1. + 0.*I

The context panel is the best quick reference for manipulation options I can think of. An easy way to learn Maple and math too. Why not adding conversion to a text book style output when Maples defaults do not provide this type of output?

Thank you all for this interesting exchange.

@acer 

I have just seen the the print extension. This is close to what I was looking for when I initially tried convert. I really think that this should be considered as a permanent extension to convert and the context panel. Really good!!!

@dharr 
Looks like. Unfortuneatly that is not what we find in text books.

And: Why a two argument function?

2*polar(1,Pi)-polar(2,Pi);
simplify(%)
                 2 polar(1, Pi) - polar(2, Pi)

                               0

Euler would turn over in his grave if he saw how wasteful we are with screen space.

But maybe an different looking inert e could be a solution.

e^(Pi*I)
                             Pi I
                            e      

alias(e=%exp): #that is not working
e(I*Pi);       # but should describe the idea I have in mind
simplify(%)

                            e(Pi I)
                            -1

@acer
I see. Automatic simplification...

I am surprised that there is nothing provided for students.
The exponential form is so important when it comes to multiplication and division of complex numbers by hand.

It would be nice to demonstrate stepwise calcualtions/simplifications with Maple.

Brilliant ideas are needed.

Thanks

@vv 

IMO it should because it converts without ICs to the expected from.

odetest((lhs-rhs)(mysol),ode);

If this was intentionally implemented it should also work with ICs.

I would report it as a product improvement.

The reduction in size is impressive.
 👍

When plotting, some smart_int results differ from Maples default output. Which one to chose?
The choice is probably less imporant in the context of ODEs since integration constants must be determined anyway

x*arctan(x)^2*ln(x^2+1);
simplify(int(%,x)-smart_int(%,x)):
plot(%,x=-Pi..Pi);

x/(-x^2+5)/(-x^2+3)^(1/2);
simplify(int(%,x)-smart_int(%,x));
plot(%,x=0..Pi);

@Carl Love 

Elegant but not fully clear to me. Can you explain the yellow part and its purpose?

I understand

type([[-2.71396014591046, 0], [-2.66663223217829, 0], [-2.66663223217829, 0.0132214378482131], [-2.71396014591046, 0.0132214378482131]],4 &under nops)

Here is what I do not

[[-2.71396014591046, 0], [-2.66663223217829, 0], [-2.66663223217829, 0.0132214378482131], [-2.71396014591046, 0.0132214378482131]] &under nops;
eval(%);# tried also value...

How should I read the code?

an expression of "type listlist(numeric) having 4 operands"

or

an expression of type listlist(numeric) and of type 4 under application of nops

I analysed:

Statistics:-TallyInto(
        A, 
        map(
            b-> b[1,1]..b[2,1],
            [indets(op(1,Q), And(listlist(numeric), 4 &under nops))[]]))

@aroche 

Thank you for the fix and the explanations. Very helpful.

@Ronan 

Yes, I wanted to give a nicer alternative for the second code line but forgot to mention that intersect has to be used

indets(eqn) minus indets(eqn,name);
% intersect indets(eqn, `^`);
rpl:=[A,B,C,`&D;`,E,F]:
Subs2:=[seq(subexpr[i]=rpl[i], i=1..nops(subexpr))]:
simplify(subs(Subs2,eqn));

@Ronan 

In my example I showed a complicated way to filter for roots and powers because I could not find this straight away:

indets(a+sqrt(b), `^`)

Looks to me that the solution Maple finds is not compatible with the IC

infolevel[dsolve]:=5;
dsolve(ode);
subs(x=0,ic,%);
%/(-2);
tan(lhs(%))=tan(rhs(%))

I remember a similar case we discussed but cannot find it right away.

@acer 

Thank you very much for the answers. Very satisfying!

@acer 

Does this mean that when I see "enter unknown" Maple refers exclusively to an anonymous procedure? In the below Maple seems to assign a "procedure without name" to the name "unknown".Is this intentional or an unfortunate coincidence?

@Carl Love 

Probably the way to go. If I understand correctly, there are command specific extenstion rules.  

?simplify,details is silent about any details in which order options are processed.

Without knowing them I cannot expect 

simplify(expr,Maple_proc1,My_proc,Maple_proc2)

to process from left to right.

Thanks for the one-liner. Always a pleasure to see those.