Using the eigenfunctions of the previous problem, written as $g _ { n } ( x )$ , the Fourier-Bessel series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } g _ { n } ( x )$ , where
A) $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$
B) $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$
C) $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$
D) $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$
E) $c _ { n } = \int _ { 0 } ^ { 2 } x ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$

Question

Quizplus · Accepted Answer

The correct formula for the coefficients in a Fourier-Bessel series involves both the function $f(x)$ and the eigenfunction $g_n(x)$ being multiplied by $x$ in the numerator of the integral expression. This accounts for the radial part of the integration in cylindrical coordinates, which is necessary when dealing with Bessel functions and their orthogonality relations. The denominator normalizes this by the integral of $x$ times the square of the eigenfunction, ensuring the series coefficients are correctly calculated.

Using the Eigenfunctions of the Previous Problem, Written As gn(x)g _ { n } ( x )gn​(x)

Using the Eigenfunctions of the Previous Problem, Written As $g _ { n } ( x )$