# An integral leads to complementary error functions

I am reading a paper Albrecher, Constantinescu and Loisel "2011Explicit ruin formulas for models with dependence among risks" and getting stuck at one integral (Example 2.4): $$\int _{\frac{\lambda }{c}}^{\infty }\frac{\lambda }{\theta c}e^{-u \theta } e^{\frac{\lambda c}{ u}} \frac{\alpha }{2\sqrt{\pi \theta ^3}}e^{-\frac{\alpha ^2}{4\theta }}d\theta.$$ The $$\theta^{-\frac{5}{2}}$$ in the integrand kills my attempts. Three authors claim that it should be: $$$$\frac{\lambda }{\alpha ^2 c}e^{-\frac{\alpha ^2c}{4\lambda }}\left\{-\frac{2 \alpha }{\sqrt{\pi \frac{\lambda }{c}}}+e^{\frac{\left(\alpha c-2 \lambda \sqrt{u}\right)^2}{4c \lambda }}\left(1+\alpha \sqrt{u}\right)\text{erfc}\left[\sqrt{\frac{u \lambda }{c}}-\frac{\alpha }{2\sqrt{\frac{\lambda }{c}}}\right]+e^{\frac{\left(\alpha c+2 \lambda \sqrt{u}\right)^2}{4c \lambda }}\left(-1+\alpha \sqrt{u}\right)\text{erfc}\left[\sqrt{\frac{u \lambda }{c}}+\frac{\alpha }{2\sqrt{\frac{\lambda }{c}}}\right]\right\},$$$$ where $$\operatorname{erfc}(t) = \frac{2}{\sqrt{\pi}} \int_t^\infty e^{-x^2} dx$$.

I cannot see how to do this. I start from the above and end with the following: $$\frac{\lambda }{\alpha c}\left\{-\frac{2 }{\sqrt{\pi \frac{\lambda }{c}}}e^{-\frac{\alpha ^2c}{4\lambda }}+\frac{ e^{\frac{u \lambda }{ c }}}{\sqrt{\pi }}\int _{\frac{\lambda }{c}}^{\infty }e^{-u \theta -\frac{\alpha ^2}{4\theta }}\text{ }\left(\frac{1}{\theta \sqrt{\theta }}+\frac{2u}{\sqrt{\theta }}\right)d\theta \right\}.$$ I cannot see why this is the first equation too. Any helps? Thanks in advance.

• Maybe you would want to simplify your problem a bit because not many MSE users will be interested after seeing the messy and huge expressions. – Szeto Dec 20 '18 at 12:23
• There is a typo, the formula with $\operatorname{erfc}$ is the correct value for $$\int_{\lambda/c}^\infty \frac \lambda {\theta c} e^{-u \theta} e^{\lambda u/c} \frac \alpha {2 \sqrt{\pi \theta^3}} e^{-\alpha^2/(4 \theta)} d\theta$$ ($e^{\lambda u/c}$ instead of $e^{\lambda c/u}$). For a given $k$, a closed form for $\int \theta^{k + 1/2} e^{-a \theta - b/\theta} d\theta$ exists because $$\int e^{-(a x + b/x)^2} dx = \frac {\sqrt \pi} {4 a} \left( e^{-4 a b} \operatorname{erfc} \left( \frac b x - a x \right) - \operatorname{erfc} \left( \frac b x + a x \right) \right).$$ – Maxim Dec 20 '18 at 14:45
• Yes，I just realise this typo. Many thanks for this. – gouwangzhangdong Dec 21 '18 at 2:46
• This solution looks very similar, but I'm not sure if it helps: math.stackexchange.com/a/3051321/441161 – Andy Walls Dec 31 '18 at 3:41