I am trying to find a way to calculate the $\phi^4$ perturbative Gaussian integral in quantum field theory without using Mathematica.
The integral is $I(\lambda)=\int_{-\infty}^{+\infty} \exp\{-x^{2}-\lambda x^{4}\}\ dx$
Mathematica shows
$I(\lambda)=\frac{e^{\frac{1}{8\lambda}}K_{\frac{1}{4}}(\frac{1}{8\lambda})}{2\sqrt{\lambda}}$, for $\Re(\lambda)>0$, where $K_{n}(x)$ is the modified Bessel function of the second kind.