I was following a lecture yesterday and we were computing the product of two independent identically distributed Gamma random variables, in the calculations we stumbled upon this integral:
$$\int_0^{+\infty} \frac{1}{x} e^{\frac{-z-x^2}{bx}} \,dx$$
and we said that with a change of variables this becomes a Bessel modified function of the second kind:
$$\int_1^{+ \infty} \frac{e^{-2 (t\sqrt{z})/b}}{\sqrt{t^2-1}} \, dt$$
I can't find this change of variables, what is it?
It seems to be:
$$\frac{z+x^2}{x}=2t\sqrt{z}$$
that is:
$$x^2-2 xt\sqrt{z}+z=0$$
$$x=\frac{2t\sqrt{z}\pm \sqrt{4t^2z-4z}}{2}=t\sqrt{z}\pm \sqrt{t^2z-z}$$
$$x=t\sqrt{z}+ \sqrt{t^2z-z}=\sqrt{z}(t+ \sqrt{t^2-1})$$
$$dx=\left(\sqrt{z}+ \frac{tz}{\sqrt{t^2z-z}}\right)dt=\left(\frac{z\sqrt{t^2-1}+zt}{\sqrt{t^2z-z}}\right)dt=\sqrt{z}\left(\frac{\sqrt{t^2-1}+t}{\sqrt{t^2-1}}\right)dt$$
$$\frac{dx}{x}=\left(\frac{1}{\sqrt{t^2-1}}\right)dt \quad \square$$