The solution to the integral $\int_{0}^{\infty} \log(x) K_{0}(2\sqrt{x})\,dx$

Question

I just found on Mathematica that the solution to the integral $$\int_{0}^{\infty} \log(x) \cdot K_{0}(2\sqrt{x})\,dx$$ is actually minus Euler–Mascheroni constant, where $K_{0}(\cdot)$ is the modified Bessel function of second kind.

This is very interesting given that $\int_{0}^{\infty} \log(x) \cdot \exp(-x)\,dx$ also equals to minus Euler constant.

However, I did not find any sources which has this equality and I am getting a little bit suspicious about this equality. I tried to use the series expansion of modified Bessel function of 2nd kind for the integral, but I still cannot get the answer, so do you guys have any brilliant ideas to prove the equality.

Thanks a lot.

Answer 1

I give you an outline that should work, and leave it to you to fill in details. The Mellin transform of $K_0$ can be calculated, $$ \int_0^{+\infty}t^{s-1}K_0(t)\,dt=2^{s-2}\Gamma(s/2)^2. $$ Your integral can after the substitution $t=2\sqrt{x}$ be written as $$ \int_0^{+\infty}t\ln (t/2) K_0(t)\,dt. $$ If we differentiate the identity from the Mellin transform with respect to $s$, and set $s=2$, we find that $$ \int_0^{+\infty} t\ln t K_0(t)\,dt=-\gamma+\ln 2. $$ Using $\int_0^{+\infty}t K_0(t)\,dt=1$, you will get

$$\int_0^{+\infty} \ln (x) K_0(2\sqrt{x})\,dx=-\gamma.$$