Chapter 8
Integral Calculus and Its Uses
8.5 Representations of Periodic Functions
Section Summary
Functions \(f\) of the form
are called trigonometric polynomials. Given such a function, we can obtain the coefficients by integration:
\(\displaystyle b_0 = \frac{1}{2 \pi} \int_{- \pi}^{\,\pi} f(t)\,dt\) |
\(\displaystyle b_k = \frac{1}{\pi} \int_{- \pi}^{\,\pi} f(t)\,\cos\,kt\,dt\) for \(k = 1, 2, ..., n\) |
\(\displaystyle a_k = \frac{1}{\pi} \int_{- \pi}^{\,\pi} f(t)\,\sin\,kt\,dt\) for \(k = 1, 2, ..., n\). |
These observations represent the first steps in the general theory of representing complicated periodic functions as combinations of simpler basic periodic functions. This representation theory is generally known as the theory of Fourier series (after Joseph Fourier). We proceed a little farther into this theory in the exercises at the end of this section.
