In the previous two problems, the infinite series solution for $u ( r , \theta )$ is $u = \sum _ { n = 1 } ^ { \infty } c _ { n } r ^ { n } \Theta _ { n } ( \theta )$ , where $\Theta _ { n }$ is found in the previous problem, and
A) $c _ { n } = 2 \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / \pi$
B) $c _ { n } = 2 \int _ { 0 } ^ { \pi } f ( \theta ) \cos ( n \theta ) d \theta / \pi$
C) $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \cos ( n \theta ) d \theta / \pi$
D) $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / \pi$
E) $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / ( 2 \pi )$

Question

Quizplus · Accepted Answer

The coefficients $c_n$ are determined by projecting the function $f(\theta)$ onto the orthogonal basis functions, which in this context are sine or cosine functions. The correct formula for $c_n$ involves an integral that captures the projection of $f(\theta)$ onto the sine functions, due to the orthogonality of sine functions over the interval $[0, \pi]$ with a normalization factor. The factor of $2/\pi$ is typical for Fourier sine series coefficients when the interval is $[0, \pi]$.

In the Previous Two Problems, the Infinite Series Solution For u