Appendix 2: Gaussian integrals

\infint e^{-y^2}dy = \sqrt{\pi}.

\begin{align*} \infint e^{-b y^2}dy &= \sqrt{\frac{\pi}{b}} \end{align*}

where $b > 0$ is a real and positive constant.

\vph\infint y e^{-b y^2}dy = 0,

where $b>0$ is a real and positive constant.

\vph \infint y^2 e^{-b y^2}dy = \frac{1}{2}\sqrt{\frac{\pi}{b^3}},

where $b>0$ is a real and positive constant.

\infint e^{-\alpha y^2 + \beta y}dy = \sqrt{\frac{\pi}{\alpha}}e^{\beta^2/4\alpha}

where α and β are complex numbers, and the real part of α is positive, $\mathrm{Re}(\alpha) > 0$ .