Evaluate an integral of the Airy function $\operatorname{Bi}(x)$

Question

How do I evaluate this interesting integral with the Airy function:

$$\int_0^x \operatorname{Bi}(u)^2 du$$

More generally, how do I evaluate

$$\int_0^x \operatorname{Bi}(u)^n du$$

Answer 1

The first one is easy. We know that $\operatorname{Bi}^{\prime\prime}(z)=z\operatorname{Bi}(z)$ from the Airy differential equation, so

$$\begin{align*} \int\operatorname{Bi}(z)^2\mathrm dz&=z\operatorname{Bi}(z)^2-2\int z\operatorname{Bi}(z)\operatorname{Bi}^\prime(z)\mathrm dz\\ &=z\operatorname{Bi}(z)^2-2\int\operatorname{Bi}^\prime(z)\operatorname{Bi}^{\prime\prime}(z)\mathrm dz\\ &=z\operatorname{Bi}(z)^2-\operatorname{Bi}^\prime(z)^2 \end{align*}$$

and then use the initial conditions for the Airy differential equation to yield

$$\int_0^z\operatorname{Bi}(t)^2\mathrm dt=z\operatorname{Bi}(z)^2-\operatorname{Bi}^\prime(z)^2+\frac{\sqrt[3]{3}}{\Gamma\left(\frac13\right)^2}$$

For $n=1$ in your more general integral, the Scorer functions and their derivatives are involved:

$$\int_0^z\operatorname{Bi}(t)\mathrm dt=\pi(\operatorname{Bi}(z)\operatorname{Hi}^\prime(t)-\operatorname{Bi}^\prime(z)\operatorname{Hi}(z))$$