Case 1
Consider the behaviour of the modified Bessel function of the second kind for $x \to 0$, when $x \in \mathbb{R}$ and $n = 0$. Abramowitz and Stegun (Handbook of Mathematical Functions, par. 9.6.8), state that
$$K_0 (x) \propto - \log(x)$$
where $\log$ is the natural logarithm. But Wolfram Alpha gives a more complete expression:
$$K_0 (x) \simeq - \log (x) - \gamma + \log (2) \ldots$$
where $\gamma$ is the Euler-Mascheroni constant. This is called a generalized Puiseux series.
Case 2
For $x \to 0$, the ratio
$$\frac{K_0 (x)}{x K_1 (x)}$$
has the same 0-th order term $- \log (x) - \gamma + \log (2)$, as $K_0 (x)$ alone, in its generalized Puiseux series expansion.
My question:
How is the number $\gamma$ generated? And how to compute this 0-th order term in both cases?