Module for Fourier Series for O. D. E.

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

Background.

    Fourier series are used to expand periodic functions in the trigonometric form.

Definition  (Fourier Series).  If is periodic with period and is piecewise continuous on, then the Fourier Series for   is

    ,

where the coefficients   are given by the so-called Euler's formulae:  

    ,  
    and  
    .  

Theorem (Fourier Expansion).  Assume that is the Fourier Series for .  If are piecewise continuous on , then is convergent for all .  The relation holds for all where f is continuous.  If is a point of discontinuity of f, then

     ,  

where denote the left-hand and right-hand limits, respectively.  With this understanding, we have the Fourier Series expansion:

     .  

Theorem (Fourier Cosine Series).  Assume that is an even function and has period .  If are piecewise continuous, the Fourier series for     involves only the cosine terms,  ,  and we write  

    ,  

where .  

Theorem (Fourier Sine Series).  Assume that is an odd function and has period .  If are piecewise continuous, the Fourier series for     involves only the sine terms,  ,  and we write  

    ,  

where  .  

Method I. Fourier Series - Trigonometric Polynomials.  Execute the following sells to set up the procedure S[n,t] for finding the Fourier Series - Trigonometric Polynomial of degree n.

Example 1.  Find the Fourier Series expansion of  .  

Method II.  Fourier Series - Trigonometric Polynomials.  

    Modify procedure S[n,t] for finding the Fourier Series - Trigonometric Polynomial of degree n to the case of a piecewise continuous function on the subintervals

Example 2.  The function    has the Fourier series representation  

    .  

Example 3.  Find the Fourier series expansion for    over ,  extended periodically with period  .  

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