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Let $X$ follow a normal distribution with mean $\mu$ and variance $\sigma^2$. Mathematica gives $$ E[\log|1+X|]=\frac{1}{2}\biggl(-\gamma-\log 2+2\log\sigma-\frac{\partial}{\partial a}{}_1 F_1\biggl(0,\frac{1}{2},-\frac{(1+\mu)^2}{2\sigma^2}\biggr)\biggr), $$ where $\gamma$ is Euler's gamma and ${}_1 F_1(a,b,z)$ the confluent hypergeometric function.

I want to know the condition on $(\mu,\sigma^2)$ under which $E[\log|1+X|]<0$. For this, I need to know how $$ \frac{\partial}{\partial a}{}_1 F_1\biggl(0,\frac{1}{2},-z\biggr) $$ behaves for $z>0$. I found this paper that derives some formula for this, but the case with $a=0$ does not seem to be covered in the expression I can understand "(6a)". Does anyone have a suggestion on how to proceed?

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    $\begingroup$ Eq. (38a) in the paper, that you linked in your question, seems to give the solution to your problem. Btw, here is a link to that paper without paywall researchgate.net/publication/… $\endgroup$
    – Johannes Trost
    Commented May 16 at 12:36
  • $\begingroup$ Thank you! I must have overlooked it. $\endgroup$
    – user108
    Commented May 17 at 12:35

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