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#197 DFLOAT format  ← ↑ →  #199 integrate(exp(-x^2)+exp(x)/x,x)

#198 integrate(sin(x**2),x) is not handled
  
last edited 10 years ago by test1

Edit detail for #198 integrate(sin(x**2),x) is not handled revision 1 of 4
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Editor:
Time: 2007/11/17 22:06:46 GMT-8
Note: Integrate 
changed:
-
Axiom doesn't seem to do the integral of sin(x^2), but both Maxima and Mathematica (per http://integrals.wolfram.com/ anyway) produce  answers.  Confirmed by Martin Rubey, and uploaded at his request to IssueTracker.

CY

\begin{axiom}
  integrate(sin(x**2),x)
\end{axiom}

However::
 
 Maxima 5.9.1.1cvs http://maxima.sourceforge.net
 Using Lisp CMU Common Lisp 19b (19B)
 Distributed under the GNU Public License. See the file COPYING.
 Dedicated to the memory of William Schelter.
 This is a development version of Maxima. The function bug_report()
 provides bug reporting information.
 (%i1) integrate(sin(x**2),x);
                                             (sqrt(2) %i + sqrt(2)) x
 (%o1) sqrt(%pi) ((sqrt(2) %i + sqrt(2)) erf(------------------------)
                                                        2
                                     (sqrt(2) %i - sqrt(2)) x
        + (sqrt(2) %i - sqrt(2)) erf(------------------------))/8
                                                2
 (%i2)

From BobMcElrath Fri Aug 19 11:13:26 -0500 2005
From: Bob McElrath
Date: Fri, 19 Aug 2005 11:13:26 -0500
Subject: (new)
Message-ID: <20050819161329.GB28462@mcelrath.org>
In-Reply-To: <20050819060559-0500@www.axiom-developer.org>

It's worth noting that both Maple and Mathematica produce the FresnelS
function, which is *defined* in terms of this integral.  However, the
Maxima answer appears to be correct as well.



From kratt6 Sat Aug 20 09:40:27 -0500 2005
From: kratt6
Date: Sat, 20 Aug 2005 09:40:27 -0500
Subject: 
Message-ID: <20050820094027-0500@page.axiom-developer.org>

Note that $sin(x^2)$ is a D-finite function:

\begin{axiom}
f := sin(x^2)
4*x^3*f - D(f,x) + x*D(f,x,2)
\end{axiom}

Thus integration should be "easy"...

From kratt6 Sat Aug 20 11:05:33 -0500 2005
From: kratt6
Date: Sat, 20 Aug 2005 11:05:33 -0500
Subject: 
Message-ID: <20050820110533-0500@page.axiom-developer.org>

I browsed the web a little more and came to the conclusion that the Risch algorithm only deals with elementary functions whose integral is elementary, too. "Clearly" (looking at maxima's output or browsing the web), $sin(x^2)$ does not have an elementary antiderivative. Hence, I suspect that it should be treated just as $e^{-x^2}$, by the 'PatternMatchIntegration' package. This would not be too difficult, probably.

Martin

From kratt6 Sat Aug 20 12:30:16 -0500 2005
From: kratt6
Date: Sat, 20 Aug 2005 12:30:16 -0500
Subject: 
Message-ID: <20050820123016-0500@page.axiom-developer.org>

Although this does not really solve the original problem, I think I found a bug in 'INTPM'. Currently, there is an operation::

           pmComplexintegrate(f, x) ==
             (rc := splitConstant(f, x)).const ^= 1 =>
               (u := pmintegrate(rc.nconst, x)) case "failed" => "failed"
               rec := u::ANS
               [rc.const * rec.special, rc.const * rec.integrand]
             cse := (rec := matcherfei(f, x, true)).which
             cse = ERF  => [rec.coeff * erf(rec.exponent), 0]
             "failed"

It is pretty obvious that the third line should read::

               (u := pmComplexintegrate(rc.nconst, x)) case "failed" => "failed"

instead. If we perform this change, we get instead of
\begin{axiom}
complexIntegrate(-%i/2*e^(%i*x^2),x)
\end{axiom}

the correct answer, same for
\begin{axiom}
complexIntegrate(%i/2*e^(-%i*x^2),x)
\end{axiom}

For some reason, it still won't do 
\begin{axiom}
complexIntegrate(sin(x^2),x)
\end{axiom}

Curiously, the pattern matcher is not even invoked in this case... Even if we enter the integral as
\begin{axiom}
complexIntegrate(-%i/2*e^(%i*x^2)+%i/2*e^(-%i*x^2),x)
\end{axiom}
it fails, although in this case the pattern matcher is invoked. It would need to be invoked on each summand seperately, though.

Martin

From kratt6 Sat Aug 20 15:56:51 -0500 2005
From: kratt6
Date: Sat, 20 Aug 2005 15:56:51 -0500
Subject: 
Message-ID: <20050820155651-0500@page.axiom-developer.org>

There is another issue I don't quite understand. Currently axiom returns the whole integral unevaluated if it does not manage to evaluate it completely. If it were not for the bug #199, the following were an example:
\begin{axiom}
integrate(exp(x)/x+exp(-x^2)+1/log(x),x)
\end{axiom}

Although axiom  produces the intermediate result 'integrate(exp(x)/x+exp(-x^2),x)+li(x)', it discards it. The responsible line of code is in 'rinteg$FSINT' ::

    rinteg(i, f, x, h, comp) ==
      not elem? i => integral(f, x)$F
      empty? rest(l := [mkPrimh(f, x, h, comp) for f in expand i]) => first l
      l

Does this make sense?

Martin

Submitted by : (unknown) at: 2007-11-17T22:06:46-08:00 (16 years ago)
Name :
Axiom Version :
Category : Severity : Status :
Optional subject :  
Optional comment :

Axiom doesn't seem to do the integral of sin(x^2), but both Maxima and Mathematica (per http://integrals.wolfram.com/ anyway) produce answers. Confirmed by Martin Rubey, and uploaded at his request to IssueTracker?.

CY

axiom
integrate(sin(x**2),x)
LatexWiki Image(1)
Type: Union(Expression Integer,...)

However: 

 Maxima 5.9.1.1cvs http://maxima.sourceforge.net
 Using Lisp CMU Common Lisp 19b (19B)
 Distributed under the GNU Public License. See the file COPYING.
 Dedicated to the memory of William Schelter.
 This is a development version of Maxima. The function bug_report()
 provides bug reporting information.
 (%i1) integrate(sin(x**2),x);
                                             (sqrt(2) %i + sqrt(2)) x
 (%o1) sqrt(%pi) ((sqrt(2) %i + sqrt(2)) erf(------------------------)
                                                        2
                                     (sqrt(2) %i - sqrt(2)) x
        + (sqrt(2) %i - sqrt(2)) erf(------------------------))/8
                                                2
 (%i2)

(new) --Bob McElrath?, Fri, 19 Aug 2005 11:13:26 -0500 reply
It's worth noting that both Maple and Mathematica produce the FresnelS? function, which is defined in terms of this integral. However, the Maxima answer appears to be correct as well.

... --kratt6, Sat, 20 Aug 2005 09:40:27 -0500 reply
Note that LatexWiki Image is a D-finite function:

axiom
f := sin(x^2)
LatexWiki Image(2)
Type: Expression Integer
axiom
4*x^3*f - D(f,x) + x*D(f,x,2)
LatexWiki Image(3)
Type: Expression Integer

Thus integration should be "easy"...

... --kratt6, Sat, 20 Aug 2005 11:05:33 -0500 reply
I browsed the web a little more and came to the conclusion that the Risch algorithm only deals with elementary functions whose integral is elementary, too. "Clearly" (looking at maxima's output or browsing the web), LatexWiki Image does not have an elementary antiderivative. Hence, I suspect that it should be treated just as LatexWiki Image, by the PatternMatchIntegration package. This would not be too difficult, probably.

Martin

... --kratt6, Sat, 20 Aug 2005 12:30:16 -0500 reply
Although this does not really solve the original problem, I think I found a bug in INTPM. Currently, there is an operation: 
           pmComplexintegrate(f, x) ==
             (rc := splitConstant(f, x)).const ^= 1 =>
               (u := pmintegrate(rc.nconst, x)) case "failed" => "failed"
               rec := u::ANS
               [rc.const * rec.special, rc.const * rec.integrand]
             cse := (rec := matcherfei(f, x, true)).which
             cse = ERF  => [rec.coeff * erf(rec.exponent), 0]
             "failed"

It is pretty obvious that the third line should read: 

               (u := pmComplexintegrate(rc.nconst, x)) case "failed" => "failed"

instead. If we perform this change, we get instead of 

axiom
complexIntegrate(-%i/2*e^(%i*x^2),x)
LatexWiki Image(4)
Type: Expression Complex Integer

the correct answer, same for 

axiom
complexIntegrate(%i/2*e^(-%i*x^2),x)
LatexWiki Image(5)
Type: Expression Complex Integer

For some reason, it still won't do 

axiom
complexIntegrate(sin(x^2),x)
LatexWiki Image(6)
Type: Expression Integer

Curiously, the pattern matcher is not even invoked in this case... Even if we enter the integral as 

axiom
complexIntegrate(-%i/2*e^(%i*x^2)+%i/2*e^(-%i*x^2),x)
LatexWiki Image(7)
Type: Expression Complex Integer

it fails, although in this case the pattern matcher is invoked. It would need to be invoked on each summand seperately, though.

Martin

... --kratt6, Sat, 20 Aug 2005 15:56:51 -0500 reply
There is another issue I don't quite understand. Currently axiom returns the whole integral unevaluated if it does not manage to evaluate it completely. If it were not for the bug #199, the following were an example: 
axiom
integrate(exp(x)/x+exp(-x^2)+1/log(x),x)
LatexWiki Image(8)
Type: Union(Expression Integer,...)

Although axiom produces the intermediate result integrate(exp(x)/x+exp(-x^2),x)+li(x), it discards it. The responsible line of code is in rinteg$FSINT : 

    rinteg(i, f, x, h, comp) ==
      not elem? i => integral(f, x)$F
      empty? rest(l := [mkPrimh(f, x, h, comp) for f in expand i]) => first l
      l

Does this make sense?

Martin