An Apparent Paradox?

[gcook-2]

I tried to double integrate 1/(1-(x*y)^2)and got a "complex" answer. I then separated the fraction into partial fractions and integrated them. This time I got a "real" answer.
I'm perplexed as to why M'Cad (my version 12)would give two differing answersthat do not appear to be equivalent
GC

[TomGutman]

You have a singularity in your integration range (at x=1, y=1). Equations tend to break down at singularities.

Your expression for the partial fraction expansion in terms of x and y is incorrect.
__________________
� � � � Tom Gutman

[AlvaroDíaz]

There are not paradox, but a bug in maple vr4 double integrating 1/(1-x^2*y^2). Correct answer is Pi^2/8.

Regards. Alvaro.

[gcook-2]

jmG; Obviously, my "Interest" is why Math cad 12 returns TWO DIFFERING results for the double definite integral of a function and its mathematically identical equivalent.

Unless the problem is associated with infinities that occur at x = y = 1, then I am led to wonder at the correctness of the Math cad solution. The partial fraction integration seems to be correct (Pi^2/8)

I've looked at your data sheet but cannot see a direct answer to my original quandary.
G.C.


On 12/23/2009 2:45:40 AM, GCOOK wrote:
>I tried to double integrate
>1/(1-(x*y)^2)and got a
>"complex" answer. I then
>separated the fraction into
>partial fractions and
>integrated them. This time I
>got a "real" answer.
>I'm perplexed as to why M'Cad
>(my version 12)would give two
>differing answers that do not
>appear to be equivalent GC
_____________________________

This integral seems to have no interest.
Who did what in the P7... work sheet ?
No matter, it is invalid.
The attached is as far as it makes sense,
The numerical integrator would fail anyway.

jmG

[ptc-1368288]

On 12/23/2009 2:45:40 AM, GCOOK wrote:
>I tried to double integrate
>1/(1-(x*y)^2)and got a
>"complex" answer. ...
>GC
__________________________

I will end it there, demonstrated and illustrated. For mortals of our kind (simple users) there is simply too much behind the scene of the CAS numerical integration, that's it. For this example, I'm convinced Maple is correct as well as Mathematica. Considering that Mathematica runs 600 pages of coding and exhausts all possible and known numerical integrations, and that it didn't report any of the blue page warning, then the numerical value is good to me and for yourself too if you wish.



jmG

[ptc-1368288]

This double integral companion:



jmG

[PhilipOakley]

Computer Algebra Systems (CAS) have to produce results according to algorithms and pre-coded tables. Decause you start in different places you are likely to end in different places, especially if you ask for different algorithms.

It is like asking to 'drive south' using main highways. where you end up depeneds on where you start. You could also try using 'the first road that goes south', etc.

In most cases we ask the CAS system for an answer because we don't know it ourselves. Trying to get get a CAS to get the answer the book gave is usually harder than herding cats.

enjoy.

Philip Oakley

[ptc-1368288]

On 1/13/2010 8:33:25 AM, philipoakley wrote:
"Computer Algebra Systems (CAS) have to produce results according to algorithms and pre-coded tables. Because you start in different places you are likely to end in different places, especially if you ask for different algorithms.

It is like asking to 'drive south' using main highways. where you end up depends on where you start. You could also try using 'the first road that goes south', etc.

In most cases we ask the CAS system for an answer because we don't know it ourselves. Trying to get get a CAS to get the answer the book gave is usually harder than herding cats.

enjoy.

Philip Oakley "
_________________________

What is your point after my tutoring/illustration/complete proof ???

"Trying to get get a CAS to get the answer the book gave is usually harder than herding cats."

No screen shot from the book, no time for that.
Just do it yourself and attach.

jmG

[ptc-1368288]

On 1/13/2010 8:33:25 AM, philipoakley wrote:
>Computer Algebra Systems (CAS)
>have to produce results
>according to algorithms and
>pre-coded tables.
...
>enjoy.
>
>Philip Oakley
___________________________

Read +

http://web.archive.org/web/20050206185937/www.cecm.sfu.ca/

jmG

[AlvaroDíaz]

On 1/13/2010 8:33:25 AM, philipoakley wrote:
>Computer Algebra Systems (CAS)
>have to produce results
>according to algorithms and
>pre-coded tables. Decause you
>start in different places you
>are likely to end in different
>places, especially if you ask
>for different algorithms.

Maple's bug isn't directly in integration procedure, which is primary based in table-search schema, but it is in the evaluation of a limit, as I try to show at the end of the worsheet that I post. Is well know that our old Maple version have a poor limit implementation.

Constrasting answers obtained via different ways is a very good engineering practice. In that sense, all path must to end in Roma.

Also, the complex maple answer isn't the supplementary complex value for the integral; that's obvious because, in the notation of my worksheet, re(I(g)) = 1/8*Pi^2+1/8, but the correct integral value is I(f) = 1/8*Pi^2.

Regards. Alvaro.