Closed Form Evaluation of

We study this integral when parameter values make it well-defined, that is to say for and .

Here again, we first input the integrand and make our choice of variables with respect to which we want to study the dependency. This has to contain the integration variable . Here, we made the choice to leave no parameter.

>

> sys[1]:=dfinite_expr_to_sys(expr,f(n::shift,z::diff,u::diff));

The integration itself takes place in the following call. The justification that we may use the option , and thus, that we may perform an integration over natural boundaries is the following. We will consider this integral for in a neighbourhood of 0, so that the exponential term is exponentially small when tends to infinity. At 0, the integrand has valuation in , so that any linear combination of the integrand and its derivatives with polynomial function coefficients will be 0 at , provided becomes sufficiently large.

> sys[2]:=int_of_sys(sys[1],z=0..infinity,takayama_algo);

The above system cancels the integral for sufficiently large . (But unfortunately, we do not know at this point any lower bound for the valid .)

At this point, we face a non-deterministic behaviour of Maple: the above output of int_of_sys varies, according to the system on which this session is run. Avoiding this non-determinism would require that the user inputs some constraints that are not expressable at the level of the Mgfun package, but only at the lower-level of the Holonomy package (this relates to choices of term orders in Groebner bases, see the Groebner package). Due to this, we provide two ways to solve the system, depending on the output of the following two Maple commands:

> map(indets,sys[2],function);

> if nops(remove(has,%,[D,diff]))=1 then "Case A" else "Case B" fi;

Please, directly enter the section with title indicated by the last output above.

Case A

Case B