If and are distinct roots of Bessel function with ,&

Question

If $k_m$ and $k_n$ are distinct roots of Bessel function $J_p(kb) = 0$ with $p\geq0$ , $b>0$ then show that $\int_{0}^{b}x J_p(k_mx)J_p(k_nx)dx = \left\{\begin{matrix} 0 & if m\neq n\\ \frac{b^2}{2}[J_{p+1}(k_nb)] & if \: m = n \end{matrix}\right.$

15 Feb 2024

Answer :

Word Count : 226

To prove the given result, we will use the orthogonality property of Bessel functions. The orthogonality property states that if $ p $ is a non-negative integer, then for distinct roots $ k_m $ and \( k_n ____ __________ _____ ____ ___.
________ _________ _______ __________ _____ ______ ________ ____ __________ ________ _______.
_____ _________ _______ __________ _______ _______.
____ ____ ________ ______ _______ ____ ________ __________ _____ _____ _________.
_________ ________ ___ ________ ___ ________.
______ _______ ___ ___ ___ _______.
_____ ____ _______ ___ ______.
_____ ______ _______ _________ _______ ________ _____ _____ ____ _______ _________ _____.
___ ____ _____ ___ _________.
__________ ________ ____ _________ ________ ___ ___ ________ _________ ______ _______.
______ _____ _______ _____ _________ __________ _______ _________ ____ ________ _________.
_______ _______ __________ _________ _________.
________ ________ ___ ____ ______ ____ ________.
___ _____ _____ _____ _____ ______ __________ ______ ____ __________ ______ _______.
_______ _____ _____ _________ _______ _____ _______ ____ _________ ________ ______.
_______ ________ ________ ____ __________ _________.
_________ _________ _______ ___ _________ __________ ___ __________.
_____ ________ _______ ____ __________.
________ _____ ______ __________ ____ ___ ________.
___ ________ _______ ______ ______ _____ __________ _______ ___ ____ __________ ____.
__________ ______ _________ _______ _________ ________ ________ __________.
_______ _____ _______ ____ ______.
__________ _________ _________ ________ __________ _____.
_________ ____ ______ _____ ______ ___.
________ ______.
Get Full Answer on WhatsApp