I am interesting in the following integral $$\int_{-\infty }^{\infty } e^{-\frac{g z^3}{6}-\frac{z^2}{2}} \, dz.$$ Mathematica does not provide any result nor maple either I try to used$$ \text{NIntegrate}\left[e^{-\frac{g z^3}{6}-\frac{z^2}{2}},\{z,-\infty ,\infty \}\right]$$ but no results. I have been able to obtain a Taylor expansion in the form of series, but it is divergent except for very small values of $g$, but I have been able to obtain values using other analytical methods. $$\sum _{k=0}^{\infty } -\frac{g^{k-1} \left(\sqrt{\pi } (-1)^k 2^{2 k-\frac{5}{2}} \left(\frac{3}{4}\right)_{k-1} \Gamma \left(k-\frac{3}{4}\right)\right)}{\Gamma \left(\frac{1}{4}\right) (1)_{k-1}}.$$ It is possible sum the above as $$\frac{2 e^{-\frac{1}{3 g^2}} \sqrt[3]{\frac{1}{g^2}} g \left(3 \pi \sqrt[3]{\frac{1}{g^2}} \sqrt[3]{-g^2} I_{\frac{1}{3}}\left(\frac{1}{3 g^2}\right)-\sqrt{3} \Gamma \left(-\frac{2}{3}\right) \Gamma \left(\frac{2}{3}\right) I_{-\frac{1}{3}}\left(\frac{1}{3 g^2}\right)\right)}{9 \sqrt[6]{-g^2}}.$$ I found using Borel sum. Now the result agree. I wanted to change the question and I am curious because all the values of the integral are complex, how is this interpreted from a physical point of view that the energy is complex?
