Intergratability test?
Chris Bennardo
Hi there,
Does anyone know if there exist a formula or procedure to determine if a
function is intergratable or not? For instance, if I have
X^2
Then all I would have to do is apply the intergratability test to X^2, get
a result, and that result would indicate that the integral of X^2 exists.
(which is, of course, (X^3)/3 )
Consider e^(x^2). If I apply this imaginary intergratability test to
e^(x^2), then I would get a value that indicates that e^(x^2) has no
closed integral.
I am NOT looking for a magic formula to give me the integral of an
equation, just a formula or procedure to tell me IF an equation is
intergratable.
Thanks,
,==,
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D Walluck
When I integrate E^(x^2) with The Integrato, I get this:
(1/2) Sqrt[Pi] Erfi[x]
whatever that means.
THE THINKER
That means an answer using an Error Function
You see e^x^2 cannot be integrated using normal integration methods.
The Thinker.
Robert Israel
In article <Pine.A32.3.95.970518...@atlas.vcu.edu>, Chris Bennardo <ted...@atlas.vcu.edu> writes:
|> Does anyone know if there exist a formula or procedure to determine if a
|> function is intergratable or not? For instance, if I have
|>
|> X^2
|>
|> Then all I would have to do is apply the intergratability test to X^2, get
|> a result, and that result would indicate that the integral of X^2 exists.
|> (which is, of course, (X^3)/3 )
|>
|> Consider e^(x^2). If I apply this imaginary intergratability test to
|> e^(x^2), then I would get a value that indicates that e^(x^2) has no |> closed integral.
The question you are asking about is called "integration in finite terms", not
"intergratability" (or even "integrability"). There is an algorithm for this,
due to Risch. In general it is not easy to carry out, although some simple
cases can be (such as your examples) can be done by hand. It is implemented
(at least partially) in most computer algebra systems such as Maple, Mathematica
and Macsyma.
For an introductory look at this subject, you might try:
A.D. Fitt & G.T.Q. Hoare, "The closed-form integration of arbitrary
functions". Mathematical Gazette (1993) 227--236.
E. Marchisotto & G. Zakeri, "An invitation to integration in finite
terms". College Math. J. 25 (1994) 295--308.
Robert Israel isr...@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4
THE THINKER
Dann Corbit wrote:
>
> THE THINKER <ah....@ic.ac.uk> wrote in article
> <33808DC8...@ic.ac.uk>...
> > D Walluck wrote:
> > >
> > > When I integrate E^(x^2) with The Integrato, I get this:
> > >
> > > (1/2) Sqrt[Pi] Erfi[x]
> > >
> > > whatever that means.
> >
> > That means an answer using an Error Function
>
> > You see e^x^2 cannot be integrated using normal integration methods.
> ;-)
I don't know whether you are trying to make a point or simpley haven't
got anything better to do.
And please don't get offended. Its just a straightforward question.
The Thinker