Chapter 5.5, Problem 10P

(a)

To determine

To show: That the $x=0$ is a regular singular point by using the given data.

(b)

To determine

To show: The roots of the indicial equation are $r_1=r_2=0$ by using the given data.

(c)

To determine

To show: That one solution for $x>0$ is $J_0\left(x\right)=1+{\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n}}{2^{2n}{\left(n!\right)}^2}}$.

(d)

To determine

To show: That the series for $J_0\left(x\right)$ converges for all x.