Summary

• A function that is periodic of period $P$ and meets certain criteria (see below) can be represented by Eq. 6.12;

  $$f(x)=\frac{A_0}{2}+ \sum_{n=1}^{\infty}A_ncos\left( \frac{n\pi x}{P/2}\right) +\sum_{n=1}^{\infty}B_nsin\left( \frac{n\pi x}{P/2}\right)$$ (6.12)

  
  
  The coefficients can be computed with
  $$A_n=\frac{2}{P}\int_{-P/2}^{P/2} f(x)cos\left( \frac{n\pi x}{P/2}\right) dx, n=0,1,2,\ldots$$
  $$B_n=\frac{2}{P}\int_{-P/2}^{P/2} f(x)sin\left( \frac{n\pi x}{P/2}\right) dx, n=1,2,3,\ldots$$
  (The limits of the integrals can be from $a$ to $a + P$)
• If $f(x)$ is an even function, only the $A$s will be nonzero. Similarly, if $f(x)$ is odd, only the $B$s will be nonzero. If $f(x)$ is neither even nor odd, its Fourier series will contain both cosine and sine terms.
• Even if $f(x)$ is not periodic, it can be represented on just the interval $(0,L)$ by redefining the function over $(-L, 0)$ by reflecting $f(x)$ about the $y$-axis or, alternative1y, about the origin. The first creates an even function, the second an odd function. The Fourier series of the redefined function will actually represent a periodic function of period $2L$ that is defined for $(-L, L)$.
• When $L$ is the half-period, the Fourier series of an even function contains only cosine terms and is called a Fourier cosine series. The $A$s can be computed by
  $$A_n=\frac{2}{L}\int_{0}^{L}f(x)cos\left( \frac{n\pi x}{L}\right) dx, n=0,1,2,\ldots$$
  The Fourier series of an odd function contain $L$s only sine terms and is called a $Fourier sine series$. The $B$s can be computed by
  $$B_n=\frac{2}{L}\int_{0}^{L}f(x)sin\left( \frac{n\pi x}{L}\right) dx, n=1,2,3,\ldots$$