# What is the distribution of the exponential of a chi-square variable?

I have a variable X that follows the chi-square distribution with one degree of freedom. Is there anything known about the distribution of $e^X$?

Given $X \sim \chi^2(1)$, let $Y=e^X$. I do not know the name of the distribution of $Y$. But I believe the density of $Y$, denoted by $p(y)$, takes the following form. Let $f(x) = \frac{1}{\sqrt{2 \pi}} x^{-1/2} e^{-x/2}$ be the density of $X$.
defined for $y>1$.