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Definition. A Fourier polynomial is an expression of the form
The Fourier polynomials are -periodic functions. Using the trigonometric identities
Using the above formulas, we can easily deduce the following result:
Theorem. Let
This theorem helps associate a Fourier series to any -periodic function.
Definition. Let f(x) be a -periodic function which is integrable on . Set
Example. Find the Fourier series of the function
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Example. Find the Fourier series of the function
Example. Find the Fourier series of the function function
Remark. We defined the Fourier series for functions which are -periodic, one would wonder how to define a similar notion for functions which are L-periodic.
Assume that f(x) is defined and integrable on the interval [-L,L]. Set
Definition. Let f(x) be a function defined and integrable on [-L,L]. The Fourier series of f(x) is
Example. Find the Fourier series of