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Topic: integral for fun
Replies: 25   Last Post: Mar 6, 2014 4:10 PM

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Richard Fateman

Posts: 1,406
Registered: 12/7/04
Re: integral for fun
Posted: Mar 1, 2014 11:30 AM
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On 3/1/2014 5:28 AM, Jonas Matuzas wrote:
....
regarding integrals in terms of elementary functions (or well-known
constants ...)

>> RJF
>
> I agree with you. You can name my integral
> Matuzas[a]=Integrate[Cos[x]^4/(a^8 + x^8), {x, -\[Infinity],
> \[Infinity]}] as new function. But you have to do table of this
> integral properties: differentiation, integration, differential
> equation it supports, Fourier series, expressions with another
> functions.


It would still not be an elementary function, even if you had
all that information available.(Historical convention, I suppose.)

If you want to learn about
the search for a solution to the problem of (indefinite) integration
in terms of elementary functions, you can try to find work
by Liouville, and then a stream of work including Robert Risch,
but not ending there. Maybe Manuel Bronstein's work would be
most recent.

And everybody should use it :) How to evaluate numerical
> -it is no problem . Actually the same is with Ci, Si and ect...
> Numerically we know how to evaluate. But the same is with elementary
> functions, is'n it?


There is a question about how to do definite integrals. Maxima
obviously used a residue method rather than substitution into
the indefinite integral. Making a wrong choice in method can
waste a lot of time.

Maxima's residue methods are described in a PhD thesis by Paul S H. Wang
at MIT.


>





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