Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1/Mistake

Source Work

 * : Chapter One: $\S 3$. Other Types of Whole-Range Series: Example $3$

Mistake

 * By formulae $(3)$ the Fourier coefficients are:
 * $a_0 = \displaystyle \int_{-1}^0 \rd x + \int_0^1 \rd x = 1$


 * $a_n = \displaystyle \int_{-1}^0 \map \cos {n \pi x} \rd x + \int_0^1 \map \cos {n \pi x} \map \cos {\pi x} \rd x = 0$


 * $b_n = \displaystyle \int_{-1}^0 \map \sin {n \pi x} \rd x + \int_0^1 \map \sin {n \pi x} \map \cos {\pi x} \rd x$


 * $= \dfrac {\map \cos {-n \pi} - 1} {n \pi} + \dfrac 1 {2 \pi} \set {\dfrac {1 - \map \cos {n + 1} \pi} {n + 1} + \dfrac {1 - \map \cos {n - 1} \pi} {n - 1} }$

Correction
The author appears to have omitted to note that the term $a_1$ is non-vanishing:

This is the graph of $\map f x$ and its expansion to $b_7$ according to the correct analysis:


 * Sneddon-1-3-Example3.png

In comparison, this is the corresponding graph as a result of 's analysis, that is, without the $a_1$ term.

The missing $a_1 \cos \pi x$ component has been presented in red, to illustrate the contribution it makes:


 * Sneddon-1-3-Example3-mistake.png