MaplePrimes - answers and comments on Question, Let's twist again

result?

Pseudomodo — Fri, 28 Oct 2011 22:54:11 Z

Is the result zero?

another way

Axel Vogt — Sun, 30 Oct 2011 19:06:15 Z

Since cos is invertible over that interval one can try:

Int(cos(a*x)*cos(x)^(a-2), x = 0 .. (1/2)*Pi);
Change(%, x=arccos(t),t);

It seems that M15 does not find an anti-derivative here?

This is one of hypotheses

Markiyan Hirnyk — Fri, 28 Oct 2011 23:11:16 Z

As far as I know, this integral equals zero under the assumptions a::posint and a>1. In other cases the numerical calculation produces a very close to zero real number. But there is a chance that the integral under consideration takes very small values depending on a.

I see.

Pseudomodo — Fri, 28 Oct 2011 23:51:07 Z

@Markiyan Hirnyk Is part of this invalid, ie. expansion of integrand, or the methods?

I'd be willing to believe that :-limit might be accidentally producing a wrong intermediate result, somewhere deep inside int. The same result is not obtained from int (for the definite integral, I mean, rather than the manual FTOC application) when :-limit is replaced by MultiSeries:-limit before the int computation.

> restart:

> int(expand(cos(a*x)*cos(x)^(a-2)),x=0..Pi/2,method=[_RETURNVERBOSE]) assuming a>1;

[ftocms = 0, ftoc = 0, FAILS = (distribution, piecewise, series, 

  o, polynomial, ln, lookup, cook, ratpoly, elliptic, 

  elliptictrig, meijergspecial, improper, asymptotic, meijerg, 

  contour)]

> AD:=int(expand(cos(a*x)*cos(x)^(a-2)),x) assuming a>1:

> limit(AD,x=Pi/2) assuming a>1;

                               0

> limit(AD,x=0) assuming a>1;

                               0
> infolevel[int]:=1: #2;

> int(expand(cos(a*x)*cos(x)^(a-2)),x=0..Pi/2,method=[ftocms]) assuming a>1;

int/indef1: first-stage indefinite integration
int/indef2: second-stage indefinite integration
int/trigon: case of integrand containing trigs
int/indef1: first-stage indefinite integration
int/indef1: first-stage indefinite integration
int/indef2: second-stage indefinite integration
int/trigon: case of integrand containing trigs
int/rischnorm: enter Risch-Norman integrator
int/rischnorm: exit Risch-Norman integrator
int/risch: enter Risch integration
int/risch: exit Risch integration
                               0

When you ask how to show it with Maple which parts are we allowed to trust for this? Are you looking for a separate analysis?

That is a help!

Markiyan Hirnyk — Sat, 29 Oct 2011 20:48:55 Z

Thank you for the interest to my question and the deep and comprehensive answer. Sorry for the delay in my reply: I was in the highlands today and just returned. Now I only vote up. I will give a detailed opinion as soon as possible. I need some time to think about this.

Opinion

Markiyan Hirnyk — Sun, 30 Oct 2011 11:22:41 Z

@Pseudomodo

I think that all three approaches are valid. The coincidence of the outputs isn't accidental. The keypoint of your solution is the usage of the expand command. This insight is difficult to invent. Now let's see a blot on the landscape. I think the method=[_RETURNVERBOSE] option is not documented so a little user can't reach it. Next, I unsuccessfully tried to verify AD:
> is(diff(AD, x) = expand(cos(a*x)*cos(x)^(a-2)));

FAIL
I didn't also succeed in this verification by hand.
The command
>int(expand(cos(a*x)*cos(x)^(a-2)),x=0..Pi/2,method=[ftocms]) assuming a>1;
doesn't work in Maple 13. I think the one works in Maple 14 or 15.
I don't understand your doubt concerning the limit command. Up to ?int/methods the method FTOCMS does the same, but uses the limit implementation in MultiSeries. This method produces the right answer. The question belongs to mathematical folklore and originates from http://en.wikipedia.org/wiki/Moscow_Institute_of_Physics_and_Technology