3. Fourier Series of Even and Odd Functions
This section can make our lives a lot easier because it reduces the work required.
In some of the problems that we encounter, the Fourier coefficients ao, an or bn become zero after integration.
Revision
Go back to Even and Odd Functions for more information.
Finding zero coefficients in such problems is time consuming and can be avoided. With knowledge of even and odd functions, a zero coefficient may be predicted without performing the integration.
Even Functions
Recall: A function y = f(t) is said to be even if f(-t) = f(t) for all values of t. The graph of an even function is always symmetrical about the y-axis (i.e. it is a mirror image).
Example of an Even Function
f(t) = 2 cos πt
Fourier Series for Even Functions
For an even function f(t), defined over the range -L to L (i.e. period = 2L), we have the following handy short cut.
Since
and
f(t) is even,
it means the integral will have value 0. (See Properties of Sine and Cosine Graphs.)
So for the Fourier Series for an even function, the coefficient bn has zero value:
bn = 0
So we only need to calculate a0 and an when finding the Fourier Series expansion for an even function f(t):
An even function has only cosine terms in its Fourier expansion:
Fourier Series for Odd Functions
Recall: A function y = f(t) is said to be odd if f(-t) = - f(t) for all values of t. The graph of an odd function is always symmetrical about the origin.
Example of an Odd Function
f(t) = sin t
Fourier Series for Odd Functions
For an odd function f(t) defined over the range -L to L (i.e. period = 2L), we find that an = 0 for all n.
Since
The zero coefficients in this case are: a0 = 0 and an = 0.
An odd function has only sine terms in its Fourier expansion.
Exercise 1
Find the Fourier Series for the function for which the graph is given by:
Exercise 2
Sketch 3 cycles of the function represented by
and f(t) = f(t + 2).
Find the Fourier Series.
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