To study the thermodynamic behavior of the limit $z\rightarrow$ 1 it is useful to get the expansions of $g_{0}\left( z\right),g_{1}\left( z\right),g_{2}\left( z\right)$
$\alpha =-\ln z$ which is small positive number. From, BE integral
$z\rightarrow 1 (\alpha\rightarrow0)$
$g_{1}\left( \alpha \right) =-ln\left( 1-z\right) =-ln\alpha+\dfrac {\alpha } {2}-\dfrac {\alpha ^{2}} {24}+O({\alpha ^{4}})$ and hence
$g_{0}\left( \alpha \right) =-\dfrac {\partial } {\partial \alpha }g_{1}\left( \alpha \right)=\dfrac {1} {\alpha }-\dfrac {1} {2}+\dfrac {\alpha } {12}O({\alpha ^{3}})$
[Source: A.Khare, Fractional Statistics and Quantum Theory, Two Dimensional Bose Gas, p.113]
Could anyone help me to derive this expressions? I can't figure out what does it mean writing this functions in the powers of ${\alpha}$.