MHB How do I separate a function into its even and odd parts?

[Dustinsfl]

There was a question but I figured it out.
$$
g(\theta) = \begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta+\pi, & -\pi\leq\theta < 0
\end{cases}
$$
So $g_e=\frac{g(\theta)+g(-\theta)}{2}$ and $g_o=\frac{g(\theta)-g(-\theta)}{2}$
\begin{alignat}{3}
g_e & = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta+\pi, & -\pi\leq\theta < 0
\end{cases}+\begin{cases}
-\theta, & 0\leq -\theta\leq\pi\\
-\theta+\pi, & -\pi\leq -\theta < 0
\end{cases}}{2}\\
& = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta+\pi, & -\pi\leq\theta < 0
\end{cases}+\begin{cases}
-\theta, & 0\geq \theta\geq -\pi\\
-\theta+\pi, & \pi\geq \theta > 0
\end{cases}}{2}\\
& = & \frac{\pi}{2}
\end{alignat}
For $g_o$, we have
\begin{alignat*}{3}
g_o & = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta + \pi, & -\pi\leq\theta < 0
\end{cases} -
\begin{cases}
-\theta, & 0\leq -\theta\leq\pi\\
-\theta + \pi, & -\pi\leq -\theta < 0
\end{cases}}{2}\\
& = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta + \pi, & -\pi\leq\theta < 0
\end{cases} +
\begin{cases}
\theta, & 0\geq \theta\geq -\pi\\
\theta - \pi, & \pi\geq \theta > 0
\end{cases}}{2}\\
& = & \theta +
\begin{cases} -\frac{\pi}{2}, & 0\leq\theta\leq\pi\\
\frac{\pi}{2}, & -\pi\leq\theta < 0
\end{cases}
\end{alignat*}

 

[HallsofIvy]

It should be emphasized that you are NOT "making" the function "even" or "odd", you are separating it into its "even" and "odd' parts.

For any function, f(x), f_e(x)=\frac{f(x)+ f(-x)}{2} is an even function, f_o(x)= \frac{f(x)- f(-x)}{2} is an odd function, and f(x)= f_e(x)+ f_o(x)