My Math Forum Even and odd periodic extensions

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 November 9th, 2016, 12:27 PM #1 Newbie   Joined: Nov 2016 From: South Africa Posts: 2 Thanks: 0 Even and odd periodic extensions Consider the piecewise function, $f(x)=\left\{\begin{array}{r|r}x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$ ---------- The odd periodic extension $F^O (x)$ of $f(x)$ is defined as: $F^O (x)=\left\{\begin{array}{r|r}0 & -3 < x < -2\\x-4 & -2 < x < 0\\x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$ where $F^O (x)$ has a period of 6. ---------- The odd periodic extension $F^O (x)$ of $f(x)$ is defined as: $F^e (x)=\left\{\begin{array}{r|r}0 & -3 < x < -2\\-x+4 & -2 < x < 0\\x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$ where $F^e (x)$ has a period of 6. ---------- Does the notation used expressions $F^O (x)$ and $F^e (x)$ represent periodic extensions of $f (x)$ that repeats indefinitely every six units? What other notation can be used to present a periodic function of a piecewise function?
 November 9th, 2016, 02:49 PM #2 Global Moderator   Joined: May 2007 Posts: 6,136 Thanks: 468 By definition a periodic expression of period 6 repeats every 6 units. The notation you use is as good as any.

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