In[21]:=

In[22]:=

Fourier Series

In this notebook we will see some examples of 2L-periodic extensions of functions. Mathematica can be used to plot them. If f(x) is the function you wish to extend, then plot f[Mod[x,2L,-L]].

Example 1

In[23]:=

In[24]:=

Here we plot the original function together with its 2-periodic extension.

In[25]:=

Out[25]=

Here is the 3-term Fourier series for f(x).

In[26]:=

Out[26]=

In[27]:=

Out[27]=

We can also plot the error in the truncated trigonometric series representation of f(x).

In[28]:=

Out[28]=

Here is the 10-term Fourier series for f(x).

In[29]:=

Out[29]=

In[30]:=

Out[30]=

We should note that where the 2-periodic extension of f(x) has a discontinuity the Fourier Series converges to 0 which is the average of -1 and 1, the right and left hand limits respectively of the periodic extension.

We can again examine the error in the approximation.

In[31]:=

Out[31]=

Example 2

In[32]:=

In[33]:=

In[34]:=

Out[34]=

Here is the 3-term Fourier series for f(x).

In[35]:=

Out[35]=

In[36]:=

Out[36]=

We can also plot the error in the truncated trigonometric series representation of f(x).

In[37]:=

Out[37]=

Here is the 10-term Fourier series for f(x).

In[38]:=

Out[38]=

In[39]:=

Out[39]=

We can again examine the error in the approximation.

In[40]:=

Out[40]=

Created by Mathematica (February 15, 2006) |