Cosine and Sine Series

Suppose/(x) is an even function defined on the interval [-ti, tt], then since cosnx is an even function and sinnx is an odd function, the product/(x)cos nx and /(x) sinnx are even and odd functions, respectively. Then the Fourier coefficients of/(x) are 

However, if a function f x) is defined only on [0, tt] then fix) may be extended in either an even extension or an odd extension. 

These functions Fe(x) and Fq(x) with period 2tt, both have Fourier series 

Following are the four examples of the expediency that is achievable when the concept of odd or even function is used in the context of Fourier series. 

---