Definite integration has been extended to handle more integrands of the following form:
where and other restrictions
on the constants apply.
In general these integrals can be expressed in terms of
the Gamma function
,
the incomplete Gamma function
,
the Polygamma function
,
the Riemann Zeta function
,
the error function,
the Fresnel integrals,
and the Meijer G function
.
Examples where
is a power of
(possibly negative and/or fractional):
> int( exp(-t^2)*cos(2*t)*t^2, t=0..infinity );
1/2
- 1/4 Pi exp(-1)
> int( exp(-sqrt(2*Pi)*t)*sin(t^2), t=0..infinity );
1/2 1/2 1/2 1/2
1/4 2 Pi - 1/2 2 Pi FresnelS(1)
> int( exp(-t)/t^(1/3), t=1..infinity );
GAMMA(2/3, 1)
> int( exp(-t)*ln(t)/t, t=1..infinity );
MeijerG(3, 0, 1)
> int( exp(-t)*ln(t)/t^(1/2), t=1..infinity );
MeijerG(3, 1/2, 1)
Maple can do some cases where the denominator of is of the
form
, e.g.
> int( exp(-t)*t^2/(1-exp(-2*t)), t=0..infinity );
7/4 Zeta(3)
And also a few cases where includes an error function, e.g.
> int( t*exp(-t^2)*erf(2*t+1), t=0..infinity );
1/2 1/2
1/2 erf(1) + 1/5 exp(-1/5) erfc(2/5 5 ) 5
> int( ln(2*t^2)*erf(t/2)/t^(3/2), t=0..infinity );
1/2 1/2 1/2
ln(2) Pi Pi Pi Psi(1/4)
6 ----------- + 8 ---------- + 2 --------------
GAMMA(3/4) GAMMA(3/4) GAMMA(3/4)