Maxima can't solve this integral equation

I think I can customize Maxima to do this automatically with matchdeclare, defrule and tellsimpafter, but it is harder than it looks.

(e.g.)

(%i1) sqrt(1-z)*sqrt(z+1)*sqrt(z^2+1);
(%o1) sqrt(1-z^4)

and 

(%i2) 1/(sqrt(1-z)*sqrt(z+1)*sqrt(z^2+1));
(%o2) 1/(sqrt(1-z^4))

Rich

 




 ------------Original Message------------
From: "Richard Hennessy"<rvh2007 at comcast.net>
To: "Barton Willis" <willisb at unk.edu>, "Raymond Toy (RT/EUS)" <raymond.toy at ericsson.com>
Cc: "Stavros Macrakis" <macrakis at alum.mit.edu>, "Raymond Toy" <toy.raymond at gmail.com>, "Maxima List" <maxima at math.utexas.edu>
Date: Fri, Jun-27-2008 7:14 PM
Subject: Re: [Maxima] Maxima can't solve this integral equation

I was wondering about that, I tried it but did not get [-1,1].  I tried to reproduce the change of variables as below but the limits of integration were not [-1,1] after I did it.  Also 

(%i1) integrate(1/sqrt(1 - -1*sin(x)^2), x, 0, %pi/2)
(eq1) integrate(1/sqrt(sin(x)^2+1),x,0,%pi/2)
(%i2) changevar(%,sin(x)^2-z^2,z,x);
`solve' is using arc-trig functions to get a solution.
Some solutions will be lost.
(eq2) integrate(1/(sqrt(1-z)*sqrt(z+1)*sqrt(z^2+1)),z,-1,0)

I believe the last output can be changed to 

(eq3) integrate(1/(sqrt(1-z^4),z,-1,0)

but Maxima will not do it.

Try to enter 

sqrt(1-z)*sqrt(z+1)*sqrt(z^2+1)
factor(%)
expand(%)
radcan(%)
ratsimp(%)
ratsimp(%),algebraic

and you will not get sqrt(1-z^4)

but I think they are the same because by trying 

sqrt(1-z^4) = sqrt(1-z)*sqrt(z+1)*sqrt(z^2+1).
at(%,[z=any old number])

the answer is always the same unless there is some special case I don't know about where they are not (like 
z = (-0)*minf.

Rich


 ------------Original Message------------
From: Barton Willis <willisb at unk.edu>
To: "Raymond Toy (RT/EUS)" <raymond.toy at ericsson.com>
Cc: "Richard Hennessy" <rvh2007 at comcast.net>, "Stavros Macrakis" <macrakis at alum.mit.edu>, "Raymond Toy" <toy.raymond at gmail.com>, "Maxima List" <maxima at math.utexas.edu>
Date: Fri, Jun-27-2008 4:48 PM
Subject: Re: [Maxima] Maxima can't solve this integral equation

-----maxima-bounces at math.utexas.edu wrote: -----

>FWIW, I think integrate(sqrt(1-z^4),z,-1,1) is 4/3*elliptic_kc(-1).

The definite integral can also be expressed in terms of gamma functions.
Unfortunately, you need to give Maxima a hint:


(%i15) integrate(sqrt(1-x^4),x,-1,1);
(%o15) integrate(sqrt(1-x^4),x,-1,1)

(%i16) 2 * integrate(sqrt(1-x^4),x,0,1);
(%o16) beta(1/4,3/2)/2

(%i17) float([float(%), 4/3*elliptic_kc(-1)]);
(%o17) [1.74803836952808,1.74803836952808]


Barton


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