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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: 1,539
Registered: 12/7/04
Re: integral for fun
Posted: Mar 4, 2014 1:38 AM
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On 3/3/2014 7:24 AM, wrote:
> On Friday, February 28, 2014 11:33:56 PM UTC-5, Richard Fateman wrote:
>> I don't consider a solution that includes
>> Si, Ci, or hypergeometric functions as a solution
>> in closed form in terms of elementary functions.
>> Unless there is no way of expressing the answer in
>> terms of elementary functions.
>> After all, you could always decide that the
>> difficult integral in question deserves its own
>> name, say FooI, and then return the answer in terms
>> of FooI.

> _Indefinite_ integral above can not be done using elementary
> functions. For such integral 'li' and 'Ei' play the same
> as logarithms. 'Ci' and 'Si' are similar to 'atan'.
> As long as CAS can compute needed limits at infinity computing
> indefinite integral in terms of special functions
> is valid method of computing definite integral.
> And it is much more general than methods based on
> residue theorem.
> Waldek Hebisch

If your goal is not to compute a result in terms of elementary
functions but to allow terms of the "higher" or "special" functions
of applied mathematics, that's fine.

You just have to draw the
line somewhere, e.g. functions in Abramowitz and Stegun. or
the NIST digital library.

One of the uses of computer algebra systems is to find
explicit formulas when possible, and it is a puzzle whether
to use more functions, e.g. Si, Ci, Li; or just express
those and other functions as hypergeometric functions.
So the idea of what is needed for an explicit solution
is somewhat fluid. Macsyma for example, generally doesn't use
division internally. a/b is really a* (b^(-1)).
A trade-off between minimizing the number of different
functions and convenience. For display purposes, Macsyma
prints a/b. (free Maxima = Macsyma essentially)

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