The Bessel functions $J_{n}(r)(n \geqslant 0)$ can be defined by the expansion

$$e^{i r \cos \theta}=J_{0}(r)+2 \sum_{n=1}^{\infty} i^{n} J_{n}(r) \cos n \theta$$

By using Cartesian coordinates $x=r \cos \theta, y=r \sin \theta$, or otherwise, show that

$$\left(\nabla^{2}+1\right) e^{i r \cos \theta}=0$$

Deduce that $J_{n}(r)$ satisfies Bessel's equation

$$\left(r^{2} \frac{d^{2}}{d r^{2}}+r \frac{d}{d r}-\left(n^{2}-r^{2}\right)\right) J_{n}(r)=0$$

By expanding the left-hand side of $(*)$ up to cubic order in $r$, derive the series expansions of $J_{0}(r), J_{1}(r), J_{2}(r)$ and $J_{3}(r)$ up to this order.