Date: Jul 24, 2013 3:14 AM Author: Nasser Abbasi Subject: Re: An independent integration test suite On 7/24/2013 1:25 AM, daly@axiom-developer.org wrote:

> On Wednesday, July 24, 2013 1:21:51 AM UTC-4, Albert Rich wrote:

>> On Tuesday, July 23, 2013 2:32:20 PM UTC-10, da...@axiom-developer.org wrote:

>>

>>

>>

>>> We differ on some results because Rubi (or whatever program you are using

>>

>>> to create the optimal results) assumes that square roots have only a single

>>

>>> positive value. Axiom does not want to make this simplification so given

>>

>>> sqrt(3)*sqrt(7)*sqrt(21) - 21

>>

>>> Axiom will not simplify this to zero but the Rubi test suite does.

>>

>>

>>

>> Surly there must be a way to tell Axiom to choose the principal branch so sqrt(4)-2 will simplify to zero?

>>

>>

>>

>> Albert

>

> Actually, as you can guess, the real results are more complicated.

> Given

>

> (sqrt(b^2*x^2-4)-sqrt(b*x-2)*sqrt(b*x+2))/

> (sqrt(b*x-2)*sqrt(b*x+2)*sqrt(b^2*x^2-4))

>

> under what conditions is this identically zero?

>

> If I extract the numerator and ask if it zero?

> Axiom replies false. Is there an algorithm to show

> that this is identically zero? I'd be happy to

> implement it. I reread a paper on the subject and

> now I'm more confused than when I started.

>

> Tim

>

hello Tim;

This is in M 9.01: For domain Reals, Mathematica

gives conditions for zero

--------------------------

Clear[x, b];

num = (Sqrt[b^2*x^2 - 4] - Sqrt[b*x - 2]*Sqrt[b*x + 2]);

den = Sqrt[b*x - 2]*Sqrt[b*x + 2]*Sqrt[b^2*x^2 - 4];

Reduce[(num/den) == 0, {x, b}, Reals]

------------------------------

(x < 0 && b < 2/x) || (x > 0 && b > 2/x)

But in Complex domain, it gives warnings due to branch cuts,

so not possible to decide on conditions for sure in this

case:

-------------------------------

Clear[x, b];

num = (Sqrt[b^2*x^2 - 4] - Sqrt[b*x - 2]*Sqrt[b*x + 2]);

den = Sqrt[b*x - 2]*Sqrt[b*x + 2]*Sqrt[b^2*x^2 - 4];

Reduce[(num/den) == 0, {x, b}, Complexes]

-------------------------------

----------------------

During evaluation of In[29]:= Reduce::useq: The answer found by

Reduce contains unsolved equation(s) {0==(-Sqrt[-2+Times[<<2>>]]

Sqrt[2+Times[<<2>>]]+Sqrt[-4+Power[<<2>>] Power[<<2>>]])/Sqrt[2+b x]}.

A likely reason for this is that the solution set depends on branch

cuts of Mathematica functions. >>

Out[32]= Sqrt[2 + b x] != 0 &&

0 == (-Sqrt[-2 + b x] Sqrt[2 + b x] +

Sqrt[-4 + b^2 x^2])/Sqrt[2 + b x] && -2 + b x != 0

------------------

But if one wants one instance which makes num/den = 0, then

there is a command for that:

FindInstance[(num/den) == 0, {x, b}, Complexes]

{{x -> 0, b -> 0}}

--Nasser