I'm trying to approximate the following Meijer-G function as $x$ goes to zero.
$G_{1,3}^{3,1}\left( {_{0,0,1}^0\left| x \right.} \right) \xrightarrow{x \rightarrow 0} \;\; \simeq ???$
To this end, I think the first step should be simplifying it. However, I'm unable to do it. I tried with both Mathematica and MuPAD of Matlab but all failed.
Can anybody give me a hint how to do it?
Thank you very much.
What you sent me in a comment seems to be $$f(x)=G_{1,3}^{3,1}\left(x\left| \begin{array}{c} 0 \\ 0,0,1 \end{array} \right.\right)$$
Using a CAS, a series expansion gives $$f(x)=-2 \gamma-\log (x) +x \left(\frac{1}{2}\log ^2(x)+(2 \gamma -1) \log (x)+\frac{\pi ^2}{3}+2 \gamma ^2-2 \gamma +1\right)+O\left(x^2\right)$$ Using $x=10^{-k}$ and computing $$\left( \begin{array}{ccc} k & \text{exact} & \text{approximation} \\ 1 & 1.703729585 & 1.757868806 \\ 2 & 3.586579192 & 3.587682919 \\ 3 & 5.779898277 & 5.779917509 \\ 4 & 8.060388203 & 8.060388503 \\ 5 & 10.35917711 & 10.35917711 \\ 6 & 12.66117633 & 12.66117633 \end{array} \right)$$