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Edit detail for #6 integration result Gamma not TeXed correctly revision 1 of 7

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Editor:
Time: 2007/12/20 11:33:13 GMT-8
Note:

changed:
-
integrate((x^2)*(%e^(-x^2)),x=0..%plusInfinity) does not give
sqrt(%pi)/4.

From BobMcElrath Wed Dec 22 21:57:46 -0600 2004
From: Bob McElrath
Date: Wed, 22 Dec 2004 21:57:46 -0600
Subject: (new)
Message-ID: <20041223035753.GD8990@mcelrath.org>
In-Reply-To: <20041222212251-0600@page.axiom-developer.org>

anonymous [mathaction@axiom-developer.org] wrote:
> integrate((x^2)*(%e^(-x^2)),x=0..%plusInfinity) does not give
> sqrt(%pi)/4.

The answer it gives is correct (unless there is a new bug introduced
that is not on my axiom 0.20040831-1).  The gamma function has sqrt(pi)
for half-integer values.

\begin{axiom}
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)
Gamma(3/2)/2::Float
sqrt(%pi)/4::Float
\end{axiom}

What I don't understand is why the integration result is so hard to
convert to a Float.  Appending '::Float' to the 'integrate()' command
fails.  Why?  If I type 'Gamma(3/2)/2' by hand it has no problem.



From BobMcElrath Wed Dec 22 22:13:50 -0600 2004
From: Bob McElrath
Date: Wed, 22 Dec 2004 22:13:50 -0600
Subject: (new)
Message-ID: <20041223041351.GE8990@mcelrath.org>
In-Reply-To: <20041223035753.GD8990@mcelrath.org>

Actually the text output is::

          _ 3
         | (-)
            2
   (28)  -----
           2
                    Type: Union(f1: OrderedCompletion Expression Integer,...)

which I assume is the Gamma function, but the tex is converted improperly::

    {{|{\overline {\ }}} 
    \left(
    {{3 \over 2}} 
    \right)}
    \over 2 
    \leqno(29)

i.e. clearly does *not* contain '\Gamma'.



From BillPage Thu Dec 23 01:32:23 -0600 2004
From: Bill Page
Date: Thu, 23 Dec 2004 01:32:23 -0600
Subject: {|{\overline {\ } == Gamma?
Message-ID: <20041223013223-0600@page.axiom-developer.org>

It is possible to convert this peculiar monogram to \Gamma for
display but I think the problem is more fundamental. Clearly AXIOM
does not have a consistent and complete representation of the Gamma
function. I think integrate should at least return a Gamma that can
be converted to Float or something of type Expression Integer like
sqrt(%pi)/4. But it seems that the current Gamma function always
agressively returns a DoubleFloat value. This doesn't seem very
useful to me for symbolic computations.

From BobMcElrath Thu Dec 23 01:42:39 -0600 2004
From: Bob McElrath
Date: Thu, 23 Dec 2004 01:42:39 -0600
Subject: {|{\overline {\ } == Gamma?
Message-ID: <20041223074250.GG8990@mcelrath.org>
In-Reply-To: <20041223013223-0600@page.axiom-developer.org>

Bill, What is the type that is returned there?  How can I get axiom to
print the entire type rather than '...'?

I think therein lies the problem.  Note that
\begin{axiom}
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)::InputForm
\end{axiom}
seems correct.

P.S. IssueTracker + axiom rules.  Having axiom right here is so
powerful...  I will have to pay more attention to IssueTracker.  I was
half thinking to strip it when (if) I fork zwiki.  Right now
IssueTracker is a big hack that can't be uninstalled...



From kratt6 Wed Jan 19 08:34:21 -0600 2005
From: kratt6
Date: Wed, 19 Jan 2005 08:34:21 -0600
Subject: this is not a math bug, only the output is not as nice as it could be
Message-ID: <20050119083421-0600@page.axiom-developer.org>

Category: Axiom Mathematics => Axiom User Interface 
Severity: normal => minor 


From BillPage Wed Jan 19 08:54:51 -0600 2005
From: Bill Page
Date: Wed, 19 Jan 2005 08:54:51 -0600
Subject: I think this is a math bug!
Message-ID: <20050119085451-0600@page.axiom-developer.org>

I disagree. There is something fundamentally wrong here. The result of
\begin{axiom}
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)
\end{axiom}

Not only prints oddly without conversion to proper LaTeX but also it
is not actually recognized "mathematically" as Gamma().

Further, there is something wrong with the implementation of Gamma()
itself because it only works numerically and is of no use symbolically.
It seems like there are two separate Gamma functions. The one produced
by the integration is "symbolic" but does not seem to be connected to
any appropriate logic.


From BillPage Wed Jan 19 09:20:12 -0600 2005
From: Bill Page
Date: Wed, 19 Jan 2005 09:20:12 -0600
Subject: this is not a mathbug, only the output is not as nice as it could be
Message-ID: <003d01c4fe3a$6c918190$6501a8c0@Asus>
In-Reply-To: <20050119083421-0600@page.axiom-developer.org>

> 
> Category: Axiom Mathematics => Axiom User Interface 
> Severity: normal => minor 
> 

    :)

This shows where your priorities are Martin! But, thanks.

Some User Interface issue is involved here, but I don't
think that it is quite correct to classify it this way.
The issue is really what symbollic computations are
possible in Axiom involving Gamma? I would say that this
is more of a deficiency in the mathmatics.


Submitted by : (unknown) at: 2007-11-17T23:01:10-08:00 (15 years ago)
Name :
Axiom Version :
Category : Severity : Status :
Optional subject :  
Optional comment :

integrate((x^2)*(%e^(-x^2)),x=0..%plusInfinity) does not give sqrt(%pi)/4.

anonymous [mathaction@axiom-developer.org]? wrote:
integrate((x^2)*(%e^(-x^2)),x=0..%plusInfinity) does not give sqrt(%pi)/4.

The answer it gives is correct (unless there is a new bug introduced that is not on my axiom 0.20040831-1). The gamma function has sqrt(pi) for half-integer values.

axiom
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)
LatexWiki Image(1)
Type: Union(f1: OrderedCompletion? Expression Integer,...)
axiom
Gamma(3/2)/2::Float
LatexWiki Image(2)
Type: DoubleFloat?
axiom
sqrt(%pi)/4::Float
LatexWiki Image(3)
Type: Expression Float

What I don't understand is why the integration result is so hard to convert to a Float. Appending ::Float to the integrate() command fails. Why? If I type Gamma(3/2)/2 by hand it has no problem.

Actually the text output is:
          _ 3
         | (-)
            2
   (28)  -----
           2
                    Type: Union(f1: OrderedCompletion Expression Integer,...)

which I assume is the Gamma function, but the tex is converted improperly:

    {{|{\overline {\ }}} 
    \left(
    {{3 \over 2}} 
    \right)}
    \over 2 
    \leqno(29)

i.e. clearly does not contain \Gamma.

{|{\overline {\ } == Gamma? --Bill Page, Thu, 23 Dec 2004 01:32:23 -0600 reply
It is possible to convert this peculiar monogram to \Gamma for display but I think the problem is more fundamental. Clearly AXIOM does not have a consistent and complete representation of the Gamma function. I think integrate should at least return a Gamma that can be converted to Float or something of type Expression Integer like sqrt(%pi)/4. But it seems that the current Gamma function always agressively returns a DoubleFloat? value. This doesn't seem very useful to me for symbolic computations.

{|{\overline {\ } == Gamma? --Bob McElrath?, Thu, 23 Dec 2004 01:42:39 -0600 reply
Bill, What is the type that is returned there? How can I get axiom to print the entire type rather than ...?

I think therein lies the problem. Note that

axiom
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)::InputForm
LatexWiki Image(4)
Type: InputForm?

seems correct.

P.S. IssueTracker? + axiom rules. Having axiom right here is so powerful... I will have to pay more attention to IssueTracker?. I was half thinking to strip it when (if) I fork zwiki. Right now IssueTracker? is a big hack that can't be uninstalled...

this is not a math bug, only the output is not as nice as it could be --kratt6, Wed, 19 Jan 2005 08:34:21 -0600 reply
Category: Axiom Mathematics => Axiom User Interface Severity: normal => minor

I think this is a math bug! --Bill Page, Wed, 19 Jan 2005 08:54:51 -0600 reply
I disagree. There is something fundamentally wrong here. The result of
axiom
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)
LatexWiki Image(5)
Type: Union(f1: OrderedCompletion? Expression Integer,...)

Not only prints oddly without conversion to proper LaTeX? but also it is not actually recognized "mathematically" as Gamma().

Further, there is something wrong with the implementation of Gamma() itself because it only works numerically and is of no use symbolically. It seems like there are two separate Gamma functions. The one produced by the integration is "symbolic" but does not seem to be connected to any appropriate logic.

6501a8c0@Asus">this is not a mathbug, only the output is not as nice as it could be --Bill Page, 6501a8c0@Asus">Wed, 19 Jan 2005 09:20:12 -0600 reply

Category: Axiom Mathematics => Axiom User Interface Severity: normal => minor

:)

This shows where your priorities are Martin! But, thanks.

Some User Interface issue is involved here, but I don't think that it is quite correct to classify it this way. The issue is really what symbollic computations are possible in Axiom involving Gamma? I would say that this is more of a deficiency in the mathmatics.