Integral of a power of the complementary error function

Question

I would like to know if is it possible to calculate analytically the following integral: $$J=\int_0^{x_0}\operatorname{erfc}(x)^kdx$$ with $k=2,3,4,...N$ where $\operatorname{erfc}(x)$ is the complementary error function. In particular, I need to know the answer (if any) for $k=4$ Thanks in advance

Answer 1

The complementary error function can't be written in terms of elementary functions.

You could rewrite it in terms of the integral of the Gaussian density, use integration by parts, and then a table for $\Phi$ (the aforementioned function).

That or rewrite it in terms of the gamma function, or in terms of the appropriate ODE and boundary conditions, and use an ODE solver to calculate the values.

In other words you might be able to rewrite it in terms of double indefinite integrals of elementary functions (using for example integration by parts), but realistically that seems like the best that can be hoped for.