Even and Odd Functions

A function is even if the following property holds for all t:
An example of an even function is shown in Figure 1.
In gneral, a function is odd if the following property holds for all t:
As an example, observe the function in Figure 2, this is an odd function:
As a result of these facts, if
a function f(t) is odd, then all of the a_m coefficients in the Fourier Series will be zero. This is because there can be no even functions in
the Fourier Series. Similarly, if the function f(t) is even, then all
of the b_n coefficients will be zero.
On the Real Fourier Series Coefficients page,
it is noted that the square function is odd, even though the property of Equation [2] does not hold. The reason
I still call this function odd, is because if the a0 (constant) coefficient is removed, then the function does
indeed become an odd function. As a result, all the a_m coefficients (m=1,2,...) are zero.
By observing whether a function is even or odd, determining the Fourier Series Coefficients can be greatly simplified.

On this page, we'll define even and odd functions, and discuss the Fourier Series properties of these functions. ## Even Functions

**f(t) = f(-t)**[Equation 1]
**Figure 1. An Even Function.**## Odd Functions

**f(t) = -f(-t)**[Equation 2]
**Figure 2. An Odd Function.**

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