Question

5) Find the Fourier series representation of () = + || in the interval − < <
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Answer 1

Answer:

of Fourier Series and Typical Examples

Baron Jean Baptiste Joseph Fourier introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related.

Baron Jean Baptiste Joseph Fourier (1768−1830)

Fig.1 Baron Jean Baptiste Joseph Fourier (1768−1830)

To consider this idea in more detail, we need to introduce some definitions and common terms.

Basic Definitions

A function is said to have period if for all Let the function has period In this case, it is enough to consider behavior of the function on the interval

Suppose that the function with period is absolutely integrable on so that the following so-called Dirichlet integral is finite:

Suppose also that the function is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).

If the conditions and are satisfied, the Fourier series for the function exists and converges to the given function (see also the Convergence of Fourier Series page about convergence conditions.)

At a discontinuity , the Fourier Series converges to

The Fourier series of the function is given by

where the Fourier coefficients and are defined by the integrals

Sometimes alternative forms of the Fourier series are used. Replacing and by the new variables