Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
I am trying to solve the following integral $$\int_0^{\infty } \int_{a-b x}^{\infty } \exp \left(-u^2\right) \, du \, dx.$$ I know it can be represented as an integral of the complementary error function, but I was wondering if there is a closed form.
May be using series expansion ?
Thank You.
From definition $$\int \exp \left(-u^2\right) \, du=\frac{ \sqrt{\pi }}{2}\, \text{erf}(u)$$ which makes $$\int_{a-b x}^{\infty } \exp \left(-u^2\right) \, du=\frac{\sqrt{\pi }}{2} \, \text{erfc}(a-b x)$$ So, $$\int \int_{a-b x}^{\infty } \exp \left(-u^2\right) \, du \, dx=\frac{\sqrt{\pi }}{2} \,\int\text{erfc}(a-b x) \,dx=-\frac{\sqrt{\pi } }{2 b}\int\text{erfc}(y)\, dy$$
Now, you can integrate by parts and get $$\int \text{erfc}(y)=y \,\text{erfc}(y)-\frac{1}{\sqrt{\pi }}\,e^{-y^2}$$
Back to $x$, we then have to consider the limit, for $t\to \infty$, of $$-\frac{e^{-a^2}+\sqrt{\pi } (-a \,\text{erf}(a-b t)-b t \,\text{erfc}(a-b t)+a \, \text{erf}(a))-e^{-(a-b t)^2}}{2 b}$$ which does not exist if $b>0$. If $b<0$, the limit should be $$\frac{\sqrt{\pi }\, a\, \text{erfc}(a)-e^{-a^2}}{2 b}$$
