The Fourier Series of a function $f$ defined on $[ - p , p ]$ is $f ( x ) = a _ { 0 } / 2 + \sum _ { n - 1 } ^ { \infty } \left( a _ { n } \cos ( n \pi x / p ) + b _ { n } \sin ( n \pi x / p ) \right)$ where Select all that apply.
A) $a _ { 0 } = \int _ { -p } ^ { p } f ( x ) d x / p$
B) $a _ { n } = \int _ { -p} ^ { p } f ( x ) \cos ( n \pi x / p ) d x / p$
C) $a _ { n } = \int _ { - p } ^ { p } f ( x ) \sin ( n \pi x / p ) d x / p$
D) $b _ { n } = \int _ { - p } ^ { p } f ( x ) \cos ( n \pi x / p ) d x / p$
E) $b _ { n } = \int _ { - p } ^ { p } f ( x ) \sin ( n \pi x / p ) d x / p$

Question

Quizplus · Accepted Answer

The coefficients $a_0$, $a_n$, and $b_n$ in the Fourier series are calculated using specific integrals. $a_0$ is the average value of the function over the interval, which is given by the integral of $f(x)$ over the interval divided by the length of the interval, hence $a_0 = \int_{-p}^{p} f(x) dx / (2p)$, not $p$ as stated in option A, making my initial selection incorrect due to a mistake in interpreting the formula. The correct formulas for $a_n$ and $b_n$ involve integrating $f(x)$ multiplied by $\cos(n\pi x/p)$ and $\sin(n\pi x/p)$, respectively, each divided by $p$, which matches options B and E correctly. Option C incorrectly associates $a_n$ with the sine function, and option D incorrectly associates $b_n$ with the cosine function.

The Fourier Series of a Function fff Defined On [−p,p][ - p , p ][−p,p]

The Fourier Series of a Function $f$ Defined On $[ - p , p ]$