Consider the Modified Bessel function of the second kind $K_0(z)$ for $z \to 0$.
Some sources (Abramowitz and Stegun, formula 9.6.8, and this page, formula 10.30.3) state that
$$K_0(z) \sim - \log z$$
That is: $K_0(z)$ is proportional to $\log(z)$. But according to Wolfram Alpa (in the "Series expansion at z = 0" section):
$$K_0(z) = \log \left( \frac{2}{\gamma z} \right) + O(z^2) = \log 2 - \log \gamma - \log z + O(z^2)$$
where $\gamma$ is the Euler-Mascheroni constant. This is a more accurate result.
How can it be obtained? I can't find a source for this.
$K_0$ can be expressed in terms of $I_0$ as follows:
$$ K_0(z)=AI_0(z)+BI_0(z)\int \frac{dx}{xI_0^2(x)}, $$ where $A=\ln 2-\gamma$ and $B=-1$ (any solution of the underlying differential equation is of that form for arbitrary $A$ and $B$). Since $$ I_0(z)=\sum_{k=0}^\infty\left(\frac{z^k}{(2k)!!}\right)^2, $$ and $I_0(0)=1$, \begin{align} K_0(z)&=I_0(z)\left(\ln 2-\gamma-\ln z+\frac{z^2}{4}+\ldots\right) \\ &=\ln 2-\gamma-\ln z+O(z^2) \end{align} when $z\searrow 0$.