Fourier Series/x over 0 to 2, x-2 over 2 to 4/Mistake

From ProofWiki
Jump to navigation Jump to search

Source Work


Mistake

Find the half-range cosine series for
$\map f x = \begin{cases} 1 , 0 < x < 2 \\ x - 2 , 2 < x < 4 \end{cases}$
for the half-range $0 < x < 4$.

... and the required series is

$\map S x = \displaystyle 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \set {1 + \frac {4 \paren {-1}^r} {\paren {2 r - 1} \pi} } x \cos \frac {\paren {2 r - 1} \pi x} 4$.


Correction

The subsequent analysis is performed for the function:

$\map f x = \begin{cases} x & : 0 < x < 2 \\ x - 2 & : 2 < x < 4 \end{cases}$


and the required series is actually:

$\map S x = \displaystyle 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \set {1 + \frac {4 \paren {-1}^r} {\paren {2 r - 1} \pi} } \cos \frac {\paren {2 r - 1} \pi x} 4$.


Sources