Calculus

Fourier Series

Even and Odd Extensions

Page 1
Problem 1
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Problems 2-4

Suppose that a function \(f\left( x \right)\) is piecewise continuous and defined on the interval \(\left[ {0,\pi } \right].\) To find its Fourier series, we first extend this function to the interval \(\left[ {-\pi,\pi } \right].\) This can be done in two ways:

  • We can construct the even extension of \(f\left( x \right):\)
    \[
    {{f_\text{even}}\left( x \right) \text{ = }}\kern0pt
    {\begin{cases}
    f\left( {-x} \right), & -\pi \le x \lt 0 \\
    f\left( {x} \right), & 0 \le x \le \pi
    \end{cases},}
    \]
  • or the odd extension of \(f\left( x \right):\)
    \[
    {{f_\text{odd}}\left( x \right) \text{ = }}\kern0pt
    {\begin{cases}
    -f\left( {-x} \right), & -\pi \le x \lt 0 \\
    f\left( {x} \right), & 0 \le x \le \pi
    \end{cases}.}
    \]

For the even function, the Fourier series is called the Fourier Cosine series and is given by

\[{{f_\text{even}}\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{a_n}\cos nx} ,}\]

where

\[
{{a_n} }={ \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\cos nxdx} ,\;\;}\kern-0.3pt
{n = 0,1,2,3, \ldots }
\]

Respectively, for the odd function, the Fourier series is called the Fourier Sine series and is given by

\[{{f_\text{odd}}\left( x \right) }={ \sum\limits_{n = 1}^\infty {{b_n}\sin nx} ,}\]

where the Fourier coefficients are

\[
{{b_n} }={ \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\sin nxdx} ,\;\;}\kern-0,3pt
{n = 1,2,3, \ldots }
\]

We can also define the Fourier Sine and Cosine series for a function with an arbitrary period \(2L.\) Let \(f\left( x \right)\) be defined on the interval \(\left[ {0,L } \right].\) Using even extension of the function to the interval \(\left[ {-L,L } \right],\) we obtain

\[{{f_\text{even}}\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{a_n}\cos \frac{{n\pi x}}{L}} ,}\]

where

\[
{{a_n} }={ \frac{2}{L}\int\limits_0^L {f\left( x \right)\cos \frac{{n\pi x}}{L}dx} ,\;\;}\kern-0.3pt
{n = 0,1,2,3, \ldots }
\]

For the odd extension, we have

\[{{f_\text{odd}}\left( x \right) }={ \sum\limits_{n = 1}^\infty {{b_n}\sin \frac{{n\pi x}}{L}} ,}\]

where the coefficients \({b_n}\) are

\[
{{b_n} }={ \frac{2}{L}\int\limits_0^L {f\left( x \right)\sin \frac{{n\pi x}}{L}dx} ,\;\;}\kern-0.3pt
{n = 1,2,3, \ldots }
\]

Solved Problems

Click on problem description to see solution.

 Example 1

Find the Fourier Cosine series of the function

\[
f\left( x \right) =
\begin{cases}
1, & 0 \le x \le d \\
0, & d \lt x \le \pi
\end{cases}.
\]

 Example 2

Find the Fourier Cosine series of the function

\[
{f\left( x \right) \text{ = }}\kern0pt
{\begin{cases}
1 – \frac{x}{d}, & 0 \le x \le d \\
0, & d \lt x \le \pi
\end{cases}.}
\]

 Example 3

Find the Fourier Sine series of the function \(f\left( x \right) = \cos x\) defined on the interval \(\left[ {0,\pi } \right].\)

 Example 4

Find the Fourier Sine series of the function \(f\left( x \right) = x\sin x,\) defined on the interval \(\left[ {0,\pi } \right].\)

Example 1.

Find the Fourier Cosine series of the function

\[
f\left( x \right) =
\begin{cases}
1, & 0 \le x \le d \\
0, & d \lt x \le \pi
\end{cases}.
\]

Solution.

We construct even extension of the given function. The corresponding Fourier series has the form:

\[f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {{a_n}\cos nx} .\]

Calculate the Fourier coefficients \({a_0}\) and \({a_n}:\)

\[{{a_0} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)dx} }={ \frac{2}{\pi }\int\limits_0^d {dx} }={ \frac{{2d}}{\pi },}\]
\[
{{a_n} }={ \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\cos nxdx} }
= {\frac{2}{\pi }\int\limits_0^d {\cos nxdx} }
= {\frac{2}{{n\pi }}\left[ {\left. {\left( {\sin nx} \right)} \right|_0^d} \right] }
= {\frac{2}{{n\pi }}\sin nd.}
\]

Thus, the Fourier series of the step function is given by

\[{f\left( x \right) = \frac{d}{\pi } }+{ \frac{2}{\pi }\sum\limits_{n = 1}^\infty {\frac{{\sin nd}}{n}\cos nx} .}\]

Graphs of the function and its Fourier approximation for \(n = 5\) and \(n = 50\) are shown in Figure \(1.\)

Fourier series of the step function

Figure 1, d = 0.5, n = 5, n = 50

Page 1
Problem 1
Page 2
Problems 2-4