I need to compute the following
\begin{equation} J(\sigma) = 1-{\int_{-\infty}^{+\infty} \frac{e^{-(\xi-\sigma^2/2)^2/2\sigma^2}}{\sqrt{2\pi}\sigma}\log [1+e^\xi]d\xi} \tag{*} \end{equation}
where $\sigma \in \mathbb{R}$. I have following questions
- What kind of function is this?
- Which numerical method I should look into for computing $J(.)$?
- Is is possible to compute $J(.)$ in Matlab directly? I tried using integral(.) but it returns me NaN values
I am not a calculus person so I have no progress to show here. Any help will be much appreciated. Thank you