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.
fricas
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)
\begin{equation}
\label{eq1}{\sqrt{\pi}}\over 4\end{equation}
Type: Union(f1: OrderedCompletion
?(Expression(Integer)),
...)
fricas
Gamma(3/2)/2::Float
\begin{equation}
\label{eq2}{0.5}\ {\Gamma \left({1.5}\right)}\end{equation}
Type: Expression(Float)
fricas
sqrt(%pi)/4::Float
\begin{equation}
\label{eq3}0.4431134627_2637900682\end{equation}
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.
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.
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
fricas
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)::InputForm
\begin{equation}
\label{eq4}\left(/ \ {\left(^\ {\left(pi \right)}\ {\left(/ \ 1 \ 2 \right)}\right)}\ 4 \right)\end{equation}
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...
Category: Axiom Mathematics => Axiom User Interface
Severity: normal => minor
I disagree. There is something fundamentally wrong here. The result of
fricas
integrate(x^2*(exp(-x^2)), x=0..%plusInfinity)
\begin{equation}
\label{eq5}{\sqrt{\pi}}\over 4\end{equation}
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.
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.
Name: #6 wrong integration result => #6 integration result Gamma not TeXed correctly
Here is a simple patch:
diff -c /home/martin/fricas/src/algebra/op.spad.pamphlet\~ /home/martin/fricas/src/algebra/op.spad.pamphlet
*** /home/martin/fricas/src/algebra/op.spad.pamphlet~ 2007-12-23 18:14:48.000000000 +0100
--- /home/martin/fricas/src/algebra/op.spad.pamphlet 2007-12-23 19:14:41.000000000 +0100
***************
*** 443,449 ****
dpi l == "%pi"::Symbol::O
dfact x == postfix("!"::Symbol::O, (ATOM(x)$Lisp => x; paren x))
dquote l == prefix(quote(first(l)::O), rest l)
! dgamma l == prefix(hconcat(super("|"::Symbol::O, "-"::Symbol::O)), l)
setDummyVar(op, n) == setProperty(op, DUMMYVAR, n pretend None)
dexp x ==
--- 443,468 ----
dpi l == "%pi"::Symbol::O
dfact x == postfix("!"::Symbol::O, (ATOM(x)$Lisp => x; paren x))
dquote l == prefix(quote(first(l)::O), rest l)
! dgamma l == prefix(super("|"::Symbol::O, "-"::Symbol::O), l)
! @
!
! It is certainly an abuse of [[OutputForm]] to produce a Gamma as done above.
! Originally, it was even worse:
! \begin{verbatim}
! dgamma l == prefix(hconcat("|"::Symbol::O, overbar(" "::Symbol::O)), l)
! \end{verbatim}
! which was TeXed to
! $$
! {|{\overline {\ }}}
! \left(
! {x}
! \right).
! $$
!
! The right thing would be to introduce Greek letters in [[OutputForm]], but that
! should be coordinated with the new mathml package.
!
! <<package COMMONOP CommonOperators>>=
setDummyVar(op, n) == setProperty(op, DUMMYVAR, n pretend None)
dexp x ==
Warning: Gamma looks slightly different in ASCII then. A better patch is discussed on the FriCAS list, 23.12.2007
Martin
Status: open => fix proposed
patch attached. Observation by Bill Page transferred to new report #408
fixed in FriCAS
Some or all expressions may not have rendered properly,
because Latex returned the following error:
This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6)
\write18 enabled.
%&-line parsing enabled.
entering extended mode
(./2223382317764768902-16.0px.tex
LaTeX2e <2005/12/01>
Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
n, swedish, ukenglish, pinyin, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/bbm/bbm.sty)
(/usr/share/texmf-texlive/tex/latex/jknapltx/mathrsfs.sty)
(/usr/share/texmf-texlive/tex/latex/base/fontenc.sty
(/usr/share/texmf-texlive/tex/latex/base/t1enc.def))
(/usr/share/texmf-texlive/tex/latex/pstricks/pstricks.sty
(/usr/share/texmf-texlive/tex/generic/pstricks/pstricks.tex
`PSTricks' v1.15 <2006/12/22> (tvz)
(/usr/share/texmf-texlive/tex/generic/pstricks/pstricks.con))
(/usr/share/texmf/tex/latex/xcolor/xcolor.sty
(/etc/texmf/tex/latex/config/color.cfg)
(/usr/share/texmf-texlive/tex/latex/graphics/dvips.def)
(/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)))
(/usr/share/texmf-texlive/tex/latex/graphics/epsfig.sty
(/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty
(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty)
(/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty
(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty)
(/etc/texmf/tex/latex/config/graphics.cfg))))
(/usr/share/texmf-texlive/tex/latex/pst-grad/pst-grad.sty
(/usr/share/texmf-texlive/tex/generic/pst-grad/pst-grad.tex
(/usr/share/texmf-texlive/tex/latex/xkeyval/pst-xkey.tex
(/usr/share/texmf-texlive/tex/latex/xkeyval/xkeyval.sty
(/usr/share/texmf-texlive/tex/latex/xkeyval/xkeyval.tex)))
`pst-plot' v1.05, 2006/11/04 (tvz,dg,hv)))
(/usr/share/texmf-texlive/tex/latex/pstricks/pst-plot.sty
(/usr/share/texmf-texlive/tex/generic/pstricks/pst-plot.tex
v97 patch 2, 1999/12/12
(/usr/share/texmf-texlive/tex/generic/multido/multido.tex
v1.41, 2004/05/18 <tvz>)))
(/usr/share/texmf-texlive/tex/latex/geometry/geometry.sty
(/usr/share/texmf-texlive/tex/xelatex/xetexconfig/geometry.cfg)
Package geometry Warning: `lmargin' and `rmargin' result in NEGATIVE (-108.405p
t).
`width' should be shortened in length.
) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `? option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty)
(/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-texlive/tex/latex/setspace/setspace.sty
Package: `setspace 6.7 <2000/12/01>
) (/usr/share/texmf-texlive/tex/generic/xypic/xy.sty
(/usr/share/texmf-texlive/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes,
docmode, (/usr/share/texmf-texlive/tex/generic/xypic/xyrecat.tex)
(/usr/share/texmf-texlive/tex/generic/xypic/xyidioms.tex)
Xy-pic version 3.7 <1999/02/16>
Copyright (c) 1991-1998 by Kristoffer H. Rose <krisrose@ens-lyon.fr>
Xy-pic is free software: see the User's Guide for details.
Loading kernel: messages; fonts; allocations: state, direction,
utility macros; pictures: \xy, positions, objects, decorations;
kernel objects: directionals, circles, text; options; algorithms: directions,
edges, connections; Xy-pic loaded))
(/usr/share/texmf-texlive/tex/generic/xypic/xyall.tex
Xy-pic option: All features v.3.3
(/usr/share/texmf-texlive/tex/generic/xypic/xycurve.tex
Xy-pic option: Curve and Spline extension v.3.7 curve, circles, loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xyframe.tex
Xy-pic option: Frame and Bracket extension v.3.7 loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xycmtip.tex
Xy-pic option: Computer Modern tip extension v.3.3
(/usr/share/texmf-texlive/tex/generic/xypic/xytips.tex
Xy-pic option: More Tips extension v.3.3 loaded) loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xyline.tex
Xy-pic option: Line styles extension v.3.6 loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xyrotate.tex
Xy-pic option: Rotate and Scale extension v.3.3 loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xycolor.tex
Xy-pic option: Colour extension v.3.3 loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xymatrix.tex
Xy-pic option: Matrix feature v.3.4 loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xyarrow.tex
Xy-pic option: Arrow and Path feature v.3.5 path, \ar, loaded)
(/usr/share/texmf-texlive/tex/generic/xypic/xygraph.tex
Xy-pic option: Graph feature v.3.7 loaded) loaded)
(/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty)
(/usr/share/texmf/tex/latex/graphviz/graphviz.sty
(/usr/share/texmf-texlive/tex/latex/psfrag/psfrag.sty))
(/usr/share/texmf/tex/latex/sagetex.sty
Writing sage input file 2223382317764768902-16.0px.sage
) (/usr/share/texmf-texlive/tex/latex/gnuplottex/gnuplottex.sty
(/usr/share/texmf-texlive/tex/latex/base/latexsym.sty)
(/usr/share/texmf-texlive/tex/latex/moreverb/moreverb.sty)
(/usr/share/texmf-texlive/tex/latex/base/ifthen.sty))
(./2223382317764768902-16.0px.aux)
(/usr/share/texmf-texlive/tex/latex/ucs/ucsencs.def)
(/usr/share/texmf-texlive/tex/latex/jknapltx/ursfs.fd)
(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd)
(/usr/share/texmf-texlive/tex/latex/base/ulasy.fd)
Package amsmath Warning: Foreign command \over;
(amsmath) \frac or \genfrac should be used instead
(amsmath) on input line 122.
[1] [2] [3]
Missing { inserted.
<to be read again>
\
l.128 \label{eq4}\left(/ \ {\left(^\
{\left(pi \right)}\ {\left(/ \ 1 \ 2 ...
Missing } inserted.
<inserted text>
}
l.128 ...ight)}\ {\left(/ \ 1 \ 2 \right)}\right
)}\ 4 \right)\end{equatio...
[4] [5] [6] [7] (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd)
LaTeX Warning: Characters dropped after `\end{verbatim}' on input line 135.
[8] (./2223382317764768902-16.0px.aux) )
(see the transcript file for additional information)
Output written on 2223382317764768902-16.0px.dvi (8 pages, 2432 bytes).
Transcript written on 2223382317764768902-16.0px.log.