10-08-2012, 12:00 PM
Fourier Series
FourierSeriesEV.pdf (Size: 90.8 KB / Downloads: 87)
When the French mathematician Joseph Fourier (1768–1830) was trying to solve a problem
in heat conduction, he needed to express a function as an infinite series of sine and
cosine functions:
Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating problems
concerning vibrating strings and astronomy.
The series in Equation 1 is called a trigonometric series or Fourier series and it turns
out that expressing a function as a Fourier series is sometimes more advantageous than
expanding it as a power series. In particular, astronomical phenomena are usually periodic,
as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms
of periodic functions.
We have derived Formulas 3, 5, and 6 assuming is a continuous function such that
Equation 2 holds and for which the term-by-term integration is legitimate. But we can still
consider the Fourier series of a wider class of functions: A piecewise continuous function
on is continuous except perhaps for a finite number of removable or jump discontinuities.
(In other words, the function has no infinite discontinuities. See Section 2.5 for a
discussion of the different types of discontinuities.)