Even and Odd Functions

On this page, we'll define even and odd functions, and discuss the Fourier Series properties of these functions. Even Functions A function is even if the following property holds for all t: f(t) = f(-t) [Equation 1] An example of an even function is shown in Figure 1. Figure 1. An Even Function. An even function is a function that has the same value at +t as it does at -t (that is, symmetric about t=0). All of the cosine functions in the Fourier Series (cos(2*pi*n*t/T) ) are even. Odd Functions In gneral, a function is odd if the following property holds for all t: f(t) = -f(-t) [Equation 2] As an example, observe the function in Figure 2, this is an odd function: Figure 2. An Odd Function. An odd function is a function that has the negative of the value at +t as it does at -t (that is, the function f(t) is the reflection of f(-t) ). All of the sine functions in the Fourier Series (sin(2*pi*n*t/T) ) are odd functions. Now, we noted that all sine functions are odd, and all cosine functions are even functions. It's not hard to show that the sum of odd functions produce an odd function, and the sum of even functions produce an even 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.