Question

In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function f_cof period 2b, the odd function f_sof period 2b, and the function f_pof period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

Short answer

The even function f_c of period 2b is the odd function f_s of period 2b is and the function f(x) on 0 , x < b is

The graph of the functions is shown below:

Step-by-step solution

Given Information.

The given function is

Meaning of Fourier series

A periodic function f(x) is expanded using the Fourier series formula in terms of an infinite sum of sines and cosines. Any periodic function or periodic signal is decomposed into the sum of a set of simple oscillating functions, mainly sines and cosines. The formula for a function's Fourier series is as follows:

Find the even function fc.

The average value of the function is a₀/2 = 1/3.

As a result, the series is

Find the odd function fs.

The value of a₀=0.

Thus, the series of the function is

Find the complete function.

The value of a₀/2 = 1/3 and the other coefficients are:

Find the value of b_n.

Thus, the series of the function is

Plotting the functions fc, fs and fp on the graph.

Now sketch the even function the odd function and the function

on the interval 0 < x < b as: