Question

(19.10) ( \int_{0¹x J_{p}(alpha x) J_{p}(beta x) d x=eft{begin{array}{ll}0 & \text { if } alpha neq beta, frac{1}{2} J_{p+1}(alpha)=\frac{1}{2} J_{p-1}(alpha)=frac{1}{2}

Answer 1

The given expression is an integral equation that has different solutions depending on whether alpha is equal to beta or not. These solutions involve Bessel functions and their derivatives.

Step-by-step explanation:

The given expression is an integral equation in the form of

(19.10) \left( \int_{{0}}^{{x}} J_p(\alpha x) J_p(\beta x) dx \right) = \begin{cases} 0 & \text{if } \alpha \\eq \beta, \\ \frac{1}{2} J_{p+1}(\alpha) = \frac{1}{2} J_{p-1}(\alpha) = \frac{1}{2} & \text{if } \alpha = \beta \end{cases}