Re: Integration Methods in PARI

[kevin lucas <lucaskevin296@gmail.com>]

It depends on the meaning of 'such'.

I should have been clearer. The PARI manual shows how to integrate the product of a sine or cosine and a non-oscillating function. I was asking about functions that could not be expressed in this way, such as a Bessel function.

On Mon, Mar 19, 2018 at 7:04 AM, Dirk Laurie <dirk.laurie@gmail.com> wrote:

2018-03-16 22:13 GMT+02:00 kevin lucas <lucaskevin296@gmail.com>:

> I made a mistake copying the integral from paper, it should have been
> intnum(x=0, +oo, x*exp(cos(x))*sin(sin(x))/(x^2+1))
> Any help or references, PARI-specific or otherwise, for integrating such oscillating integrals are welcome.

It depends on the meaning of 'such'.

This particular integral is of the form

intnum(x=0, +oo, f(x)*w(x))

f(x) = exp(cos(x))*sin(sin(x))
w(x) = x/(x^2+1)

That is to say, the integral is the product of a periodic function and
a function that tends to <ero, If either factor is replaced by 1, the
integral does not exist.

I would go so far to say that there are no general methods that do not
exploit additional information about f(x) or w(x). The alternating
series formed by integrals over half-periods is only conditionally
convergent, which makes summation methods based on them invalid.