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

  • Thread starter Thread starter Dustinsfl
  • Start date Start date
  • Tags Tags
    even Function
AI Thread Summary
The discussion revolves around separating a function into its even and odd parts using the example of the function g(θ). The even part, g_e, is calculated as the average of g(θ) and g(-θ), resulting in a constant value of π/2. The odd part, g_o, is derived from the difference between g(θ) and g(-θ), leading to a piecewise function dependent on the interval of θ. It is clarified that this process does not change the function into even or odd forms but rather identifies and separates these components. The fundamental formulas for even and odd functions are reiterated for general application.
Dustinsfl
Messages
2,217
Reaction score
5
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*}
 
Mathematics news on Phys.org
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)
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.

Similar threads

Back
Top