Scott

The  commands ' fourier', 'fouriercos' and ' fouriersin' in the package 'inttrans' compute "Fourier transforms", which you apply to aperiodic functions. For periodic functions you get a Fourier series.

I cannot understand why, after so many years of development, Maple is not distributed with a package for Fourier series , which are more basic or elementary than Fourier transforms.

Scott

The  commands ' fourier', 'fouriercos' and ' fouriersin' in the package 'inttrans' compute "Fourier transforms", which you apply to aperiodic functions. For periodic functions you get a Fourier series.

I cannot understand why, after so many years of development, Maple is not distributed with a package for Fourier series , which are more basic or elementary than Fourier transforms.

for z=1 is:

A1 := hypergeom([-1/2, 3/8, 169/88], [81/88, 19/8], z);
S:=convert(A1,FormalPowerSeries);
S1:=eval(S,z=1);

S1 := Sum(-pochhammer(3/8,k)*pochhammer(169/88,k)*4^(-k)/k!^2/pochhammer(81/88,k)/pochhammer(19/8,k)*(2*k)!/(2*k-1),k = 0 .. infinity);

value(S1);


                                    3/2
                  11              Pi
                  -- -------------------------------
                  54                           3 Pi
                     GAMMA(5/8) GAMMA(7/8) sin(----)
                                                8

which may be rewritten as 11/54*GAMMA(4/8)*GAMMA(3/8)/GAMMA(7/8) using the reflection formula (by hand).

May be that z=1 is a better point for ICs of the ODE method?

More important would be to know whether these things can be made consistent. And if they can, they will.

Comparison of tracing by Maple  8 and Maple 12 shows that both get the same up to close to the end  of the calculation where, apparently, 'simplify/hypergeom/baseHG' produces a different result.

Restricting to this case,  a "computable function" would be completelly specified by the sets A, B and the map, something like (A,B,exp(x)-13) where x is understood as an element of  A, isn't it?

One difference that I observe tracing Maple V R5.1 and Maple 12 with this integral is that the latter, apparently, does not try indefinite algebraic integration. It can be done by hand:

Int(sqrt(t^5+6*t)*(5*t^4+6),t=0..1):
subs(_X=t,`int/algebraic2/algebraic`(subs(t=_X,op(1,%))));


                               5       3/2
                           2 (t  + 6 t)
                           ---------------
                                  3

And then evaluate at the limits. I wonder if there is something wrong with this subroutine. Ie why it is not used?

With so many integration methods to choose, an option to tell: "try this one" could be more useful than expect a fully automated algorithm to catch it.

One difference that I observe tracing Maple V R5.1 and Maple 12 with this integral is that the latter, apparently, does not try indefinite algebraic integration. It can be done by hand:

Int(sqrt(t^5+6*t)*(5*t^4+6),t=0..1):
subs(_X=t,`int/algebraic2/algebraic`(subs(t=_X,op(1,%))));


                               5       3/2
                           2 (t  + 6 t)
                           ---------------
                                  3

And then evaluate at the limits. I wonder if there is something wrong with this subroutine. Ie why it is not used?

With so many integration methods to choose, an option to tell: "try this one" could be more useful than expect a fully automated algorithm to catch it.

overlaps the right column, at least here. So, it becomes difficult to read it. Could you shorten the lines or use normal text?

Maple 9.03 outputs these hypergeometrics. Up to Maple 8 the output was 14/3*sqrt(7).

Maple 9.03 outputs these hypergeometrics. Up to Maple 8 the output was 14/3*sqrt(7).

Yes, ?add states:

Note also that the index variable i is private to the add or mul invocation. Any local or global variable of the same name visible at the time when the add or mul is executed will not be affected by it.

But why the for-loop cannot be changed to use a local index also? It would be better to adapt the software to the way the users think than adapt the users to think the way the developers did it.

sherazat, your questions should be better in a new  thread.

In Maple 12 you can find the regions (cells) on the parametric space (a,b) with different number of real roots by:

with(RootFinding[Parametric]):
eq1 := y^3-a*y^2+y-a*b=0;
m:=CellDecomposition([eq1],[y],[a,b]);

The fastest way may be make plots with your constraints and a bit larger area for reference:

CellPlot(m,samplepoints=true,view=[-5..5,-2..2]);
CellPlot(m,samplepoints=true,view=[sqrt(3)..5,0..1]);  

Then you see that the cells with  a=sqrt(3)..infinity and b=0..1 are 6..9, and you get the number of real solutions within each of them by:

for i from 6 to 9 do
NumberOfSolutions(m)[i];
end do;

                                   [6, 1]
                                   [7, 3]
                                   [8, 1]
                                   [9, 1]

The borders of these regions are given by:

for i from 6 to 9 do
NumberOfSolutions(m)[i],CellDescription(m,i);
end do;

On the borders of the region 7 (blue) there are two real solutions (one double root).

The point a=sqrt(3),b=1/9 (where the borders join) is special as there is a triple root:

subs(a=sqrt(3),b=1/9,lhs(eq1));
factor(%);
                                                3
                              1  /        (1/2)\ 
                            - -- \-3 y + 3     / 
                              27                 

sherazat, your questions should be better in a new  thread.

In Maple 12 you can find the regions (cells) on the parametric space (a,b) with different number of real roots by:

with(RootFinding[Parametric]):
eq1 := y^3-a*y^2+y-a*b=0;
m:=CellDecomposition([eq1],[y],[a,b]);

The fastest way may be make plots with your constraints and a bit larger area for reference:

CellPlot(m,samplepoints=true,view=[-5..5,-2..2]);
CellPlot(m,samplepoints=true,view=[sqrt(3)..5,0..1]);  

Then you see that the cells with  a=sqrt(3)..infinity and b=0..1 are 6..9, and you get the number of real solutions within each of them by:

for i from 6 to 9 do
NumberOfSolutions(m)[i];
end do;

                                   [6, 1]
                                   [7, 3]
                                   [8, 1]
                                   [9, 1]

The borders of these regions are given by:

for i from 6 to 9 do
NumberOfSolutions(m)[i],CellDescription(m,i);
end do;

On the borders of the region 7 (blue) there are two real solutions (one double root).

The point a=sqrt(3),b=1/9 (where the borders join) is special as there is a triple root:

subs(a=sqrt(3),b=1/9,lhs(eq1));
factor(%);
                                                3
                              1  /        (1/2)\ 
                            - -- \-3 y + 3     / 
                              27                 

It would help a lot to avoid confusions if the help pages were written in a uniform style: calling conventions, terminology, etc.

Also, I do not see in ?with a mention to subpackages. In particular, showing the difference in syntax between importing a command in a package and a subpackage.