# two integrals and a question about prime trinomials

Alec Mihailovs alec at mihailovs.com
Fri Feb 25 21:28:31 CET 2005

```----- Original Message -----
From: "Karol PENSON" <penson at lptl.jussieu.fr>

>       The integral , in Maple notation,   int(
> exp(-2*x^2)*(erf(x))^2,x=0..infinity) is a special case of a
>       more general integral :
>       int( exp(-p*x^2)*erf(b*x)*erf(c*x), x=0..infinity) =
> (1/sqrt(Pi*p))*arctan(b*c/sqrt(p*(b^2+c^2+p)),
>        see: A.P.Prudnikov, Yu.A.Brychkov and O.I.Marichev , Integrals and
> Series, vol.2 - Special Functions,
>       (Gordon & Breach, 1998)  , integral no. 2.8.19.8 ,  p.123, where
> also restrictions on p, b and c
>       can be found.
>        Alec, I have tried this integral with Maple 9 and obtained nothing.
>        Which version of Maple do you use ?

I used Maple 10. It can do this more general integral, too,

int( exp(-p*x^2)*erf(a*x)*erf(b*x), x=0..infinity) assuming p>0;

a b
arctan(---------------------)
2    2     1/2  1/2
(a  + b  + p)    p
-----------------------------
1/2
(Pi p)
Alec Mihailovs
http://math.tntech.edu/alec/

```