Recall from Example 1 that Jn(k,x) satisfies the differential equation x2y" + xy'...

Recall from Example 1 that Jn(k,x) satisfies the differential equation x2y" + xy' + (k2x2 − n2)y = 0 or, equivalently,

Let the x interval be 0 <x<c, and suppose that k is chosen so that Jn(kc) = 0; i.e., kc is any of the zeros of Jn(x) = 0. The purpose of this exercise is to derive the formula

which will be be needed when we show how to use the Sturm-Liouville theory to expand functions on 0 < x < c in terms of Bessel functions. In turn, that concept will be needed later in our study of partial differential equations. To derive (7.2), we suggest the following steps.

Thus, show that (7.3) reduces to