Sauda, c~oes,

Acabo de receber a seguinte resposta
do prof. Rousseau:

Dear Luis:

 The work by Risch in question dates from 1969 (Trans. AMS 139 (1969),
167-189 and Bull. AMS 76 (1970), 605-608).  What little I know about
the subject comes from a a Monthly article by Rosenlicht ("Integration
in Finite Terms", AMM 79 (1972), 964-972).  My belief (perhaps wrong)
is that the integration routines of Maple and Mathematica basically
implement some form of the Risch algorithm - which, given a function
in elementary form (computable by a finite number steps from
algebraic, exponential, logarithmic, trigonometric functions) decides in
a finite number of steps whether or nor the antiderivative is
elementary.  Of course, Maple goes beyond this, so it can also
represent certain antiderivatives in terms of higher transcendental
functions (e.g. erf(x)).  But if I give Maple an indefinite integral
of an elementary function and it gives no result, I take it to mean
that there is no such function in elementary terms.  Perhaps I am
giving Maple more credit than it deserves, but that is my assumption.

Now definite integrals and sums of infinite series are a whole new
ball game.  In any case, I would be interested in learning what
you find out about this series.

Have a good weekend.




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