Chapter 12.23, Problem 19MP

(a) The generating function for Bessel functions of integral order $p=n$ is $\Phi (x,h)=e^{(1/2)x\left(h-h^{-1}\right)}={\displaystyle \underset{n=-\infty }{\overset{\infty }{\sum }}{h}^n}{J}_n(x)$.

By expanding the exponential in powers of $x\left(h-h^{-1}\right)$ show that the $n=0$ term is $J_0(x)$ as claimed.

(b) Show that $x^2\frac{{\partial }^2\Phi }{\partial x^2}+x\frac{\partial \Phi }{\partial x}+x^2\Phi -{\left(h\frac{\partial }{\partial h}\right)}^2\Phi =0$.

Use this result and $\Phi (x,h)={\displaystyle \underset{n=-\infty }{\overset{\infty }{\sum }}{h}^n}{J}_n(x)$ to show that the functions $J_n(x)$ satisfy Bessel’s equation. By considering the terms in $h^n$ in the expansion of $e^{(1/2)x\left(h-h^{-1}\right)}$ in part (a), show that the coefficient of $h^n$ is a series startingwith the term $(1/n!){(x/2)}^n.$ ( You have then proved that the functions called $J_n(x)$ in the expansion of $\Phi (x,h)$ are indeed the Bessel functions of integral order previously defined by $(12.9)\text{ and }(13.1)\text{ with }p=n.)$