I encounter the following form of integrals and I would like some suggestions on how I can solve it analytically (numerically is straightforward):
$$ F(a,b,c) = \int_{0}^\beta \int_{0}^\beta \int_{0}^\beta e^{ta+t_1 b + t_2 c}~\theta(t>t_1>t_2>0)\,dt_2\, dt_1\,dt,$$
where $a,b,c,$ are just some numerical values (treat them as irrelevant constants).
Thank you for your help!
The integral separates out, via Dylan's suggestion, into the following: \begin{align*}F(a,b,c) &= \int_{0}^\beta \int_{t_2}^\beta \int_{t_1}^\beta e^{ta+t_1 b + t_2 c}\,dt\, dt_1\,dt_2 \\ &=\int_0^{\beta}e^{ct_2}\int_{t_2}^{\beta}e^{bt_1}\int_{t_1}^{\beta}e^{at}\,dt\,dt_1\,dt_2 \\ &=\int_0^{\beta}e^{ct_2}\int_{t_2}^{\beta}e^{bt_1}\left(\frac{e^{at}}{a}\right)\Bigg|_{t_1}^{\beta}\,dt_1\,dt_2 \\ &=\int_0^{\beta}e^{ct_2}\int_{t_2}^{\beta}e^{bt_1}\left(\frac{e^{a\beta}-e^{at_1}}{a}\right)\,dt_1\,dt_2 \\ &=\int_0^{\beta}e^{ct_2}\int_{t_2}^{\beta}\left(\frac{e^{a\beta+bt_1}-e^{(a+b)t_1}}{a}\right)\,dt_1\,dt_2. \end{align*} Surely you can finish? It's a bit tedious, but quite straight-forward.