The Airy function is a solution to the differential equation
\begin{align}y''(x)+x\, y(x)=0.\end{align}
Using methods from complex analysis to find solutions to ODEs in terms of contour integrals, it is relatively easy to show that one solution to this equation is given by the integral
\begin{align}y_1(x) = C \int_{\gamma_1} \, dt\, \exp\left(t^3/3 -x\,t \right),\end{align}
where $\gamma_1$ is a path beginning and ending at $\mp \frac{5 \pi}{6} \infty$, respectively, and $C$ is some real integration constant. It is convention to name this integral solution the Airy function:
\begin{align}Ai(x)= \dfrac{1}{2\pi i} \int_{\gamma_1} \,dt\, \exp\left(t^3/3 -x\,t \right),\end{align}
i.e. to choose $C=(2 \pi i)^{-1}$. Why is this?
My first guess is that we want to normalise the Airy function so that the integral over the real line $\int_{-\infty}^{\infty} dx\; Ai(x) $ is equal to 1. However I cannot calculate this integral easily and so I cannot proceed.
So two questions: why the prefactor? How to do the integral over the real line?