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#59 make install sets the wrong shell variable in the axiom script ←	↑	→ #60 bugs duplicate functions exists. optimizer complains during rebuild

#6 integration result Gamma not TeXed correctly

Edit detail for #6 integration result Gamma not TeXed correctly revision 2 of 7

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Editor: kratt6 Time: 2007/12/20 11:33:14 GMT-8
Note:

added:

From kratt6 Thu Dec 20 11:33:13 -0800 2007
From: kratt6
Date: Thu, 20 Dec 2007 11:33:13 -0800
Subject: 
Message-ID: <20071220113313-0800@axiom-wiki.newsynthesis.org>

Name: '#6 wrong integration result' => '#6 integration result Gamma not TeXed correctly'

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

(new) --Bob McElrath?, Wed, 22 Dec 2004 21:57:46 -0600 reply

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)

(1)

Type: Union(f1: OrderedCompletion? Expression Integer,...)

axiom
Gamma(3/2)/2::Float

(2)

Type: DoubleFloat?

axiom
sqrt(%pi)/4::Float

(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.

(new) --Bob McElrath?, Wed, 22 Dec 2004 22:13:50 -0600 reply

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

(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)

(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.

... --kratt6, Thu, 20 Dec 2007 11:33:13 -0800 reply

Name: #6 wrong integration result => #6 integration result Gamma not TeXed correctly