Even and Odd functions
$\fcolorbox{white}{yellow}{$f(x)$ is even if $f(-x)=f(x)$, i.e., symmetric along $y$-axis.}$
$\fcolorbox{white}{yellow}{$f(x)$ is odd if $f(-x)=-f(x)$, i.e., symmetric with respect to origin.}$
$\mm$ $\int_{-l}^l f(x)dx =
\left\{\begin{array}{ll} 0 & \text{ if $f(x)$ is odd}\\ 2\int_{0}^l f(x)dx & \text{ if $f(x)$ is even} \end{array}\right.$
In particular,
$\mm$ $\fcolorbox{white}{yellow}{Sine Series:}$ If $f(x)$ is odd, then $a_n=0$, $b_n = \frac2l \int_0^l f(x)\sin \frac{n\pi x}l dx$
$\mm$ $\fcolorbox{white}{yellow}{Cosine Series:}$ If $f(x)$ is even, then $b_n=0$, $a_n = \frac2l \int_0^l f(x)\cos \frac{n\pi x}l dx$
Product: $\mm$
(odd)$\cdot$(even) $=$ (odd); $\mm$ (even)$\cdot$(even) $=$ (even); $\mm$ (odd)$\cdot$(odd) $=$ (even);