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September 19th, 2013, 02:56 AM   #1
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Integral that cannot be written in term of elem function

Hello,

I was thinking about how to prove that a converging integral cannot be written in term of elementary functions or even in term of Ei, Meijer-J or other functions??

Do you have an idea?
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September 19th, 2013, 04:02 AM   #2
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Re: Integral that cannot be written in term of elem function

One cannot prove that on a general manner. On the contrary, one could argue : Any converging integral can be written in term of special function defined for the porpose.
Also, generally, the integral can by written in term of infinite series, i.e. the combination of an infinite number of elementary functions.
The question is : how to prove that a converging integral cannot be written in term of a combination of a finite number of elementary functions and/or referenced special functions. But what is a referenced special function ? New special functions appear which are referenced in some specialized publications, but not everywhere. There is no exhaustive list of special functions.
As a consequence, to any question such as "how to prove that a converging integral cannot be written in term of a combination of a finite number of elementary functions and/or referenced special functions", the list of special functions considered has to be given.
Now, suppose that all these specifications be given whitout ambiguity. The question remains how to pove it ?
Today, it is an open subject. Some attempts have been made only in limited cases ( list of special functions very limited).
So I think that nowadays one cannot answer to your question. To such a question the experts can give an opinion and advices, but no proof.
It is a very interesting subject, on systematic and logic wiewpoint.
A paper for popularization about the use of special functions to express integrals and more :
"Safari in the Contry of Special Functions" :
http://www.scribd.com/JJacquelin/documents
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September 19th, 2013, 03:18 PM   #3
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Re: Integral that cannot be written in term of elem function

Thank you so much Jean for your clear and interesting reply and link.

What do you think of the integral below?

(Exp[-(a*((x)^2))-(b/((x)^2))])/(((x)^2)+Alpha) dx [to be calculated from 0 to +Infinity]

The term (x^2)+Alpha is outside the integral (See image in the attachment)

I tried different calculus techniques: Integration by part/variable substitution/combination of primitives in addition to special functions like Ei and Meijer-J, but I could'nt find a close form in terms of elementary and/or special functions.

Do you have any other idea about this integral or should I define it as my own special function ?
Attached Images
File Type: gif mimetex.gif (533 Bytes, 212 views)
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September 19th, 2013, 09:53 PM   #4
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Re: Integral that cannot be written in term of elem function

Hi !
What exactly is que problem ?
In the wording :
Exp[-(a*((x)^2))-(b/((x)^2))])/(((x)^2)+Alpha) dx [to be calculated from 0 to +Infinity]
In attachment :
Exp[a*x+b/x)/(((x)^2)+Alpha) dx
No squares in the exponential ? No limits for the integral ?
That is very different, especialy if the integral is definite instead of indefinite.
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September 20th, 2013, 01:27 AM   #5
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Re: Integral that cannot be written in term of elem function

Oh!! I'm so sorry. I did a mistake. What I wanted to write is:

Exp[-(a*((x)^2))-(b/((x)^2))])/(((x)^2)+Alpha) dx which means that there is squares in the exponential. The integral expression in the attachment is wrong.

The integral has a limit because the function is borned with a function (Exp[-(a*((x)^2))-(b/((x)^2))])/(x^2) whose integral has a limit.

But I couldn't find a closed form interms of elementary/special functions!! What do you think?
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September 20th, 2013, 05:37 AM   #6
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Re: Integral that cannot be written in term of elem function

Hi !
Sometimes, it is possible to argue that an integral including parameters probably cannot be expressed on a closed form and to stengthen the argument on an indirect manner:
Hypothesis : Suppose that the integral could be expressed on a closed form.
Then, for any value of the parameters, the closed form continue to be written.
If we observe that, for some values of the parameters it is possible to compute directly a closed form, but for other values of the parameters the integral is reduced to a simpler integral which we already know that there is probably no closed form, then that is in contradiction with the first hypothesis. So, we conclude that the integral probably cannot be expressed on a closed form in general. ( but can be in some particular cases, depending on particular values of parameters).
An example is shown in attachment, allowing to draw a conclusion about your integral.
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File Type: jpg indirect proof.JPG (52.3 KB, 184 views)
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September 23rd, 2013, 01:46 AM   #7
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Re: Integral that cannot be written in term of elem function

Hello,

Thank you so much Jean for your answer. Your approach is very interesting, the document you shared with me too. I'm now convinced that the integral cannot be written in terms of elementary or special functions. I tried to calculate (using scilab software) the integral for fixed values of a=b=-1 and Alpha=1 and Between 0 and 1, so I've got the numerical result. But Between 0 and +infinity I did not get the result. Do you think Matlab CAN do it better? Is Matlab stronger enough with Laplace transforms?

Thanks
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September 23rd, 2013, 02:15 AM   #8
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Re: Integral that cannot be written in term of elem function

Hi !

I don't know about Matlab, but have a look at the WolframAlpha's result :
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File Type: jpg Wolfram.jpg (89.1 KB, 163 views)
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October 23rd, 2013, 06:39 AM   #9
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Re: Integral that cannot be written in term of elem function

Thank you so much for your help.
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