Chapter 12.19, Problem 1P

Prove equation (19.10) in the following way. First note that (19.2) and (19.3) and therefore (19.7) hold whether $\alpha$ and $\beta$ are zeros of $J_p(x)$ or not. Let $\alpha$ be a zero, but let $\beta$ be just any number. From (19.7) show that then ${\displaystyle {\int }_0^{1}x}uvdx=\frac{J_p(\beta )\alpha J_p^{\prime }(\alpha )}{{\beta }^2-{\alpha }^2}.$

Now let $\beta \to \alpha$ and evaluate the indeterminate form by L'HĂ´pital’s rule (that is, differentiate numerator and denominator with respect to $\beta$ and let $\beta \to \alpha$). Hence find ${\displaystyle {\int }_0^{1}x}uvdx=\frac{1}{2}{J}_p^{\prime 2}(\alpha )$ for $\alpha =\beta ,$ that is, for $u=v=J_p(\alpha x)$ as in (19.10). Use equations (15.3) to (15.5) to show that the other two expressions given in (19.10) are equivalent.