s

Example 12.2.  The function  [Graphics:Images/FourierSeriesComplexMod_gr_219.gif],  extended periodically by the equation  [Graphics:Images/FourierSeriesComplexMod_gr_220.gif],  has the Fourier series expansion  

        [Graphics:Images/FourierSeriesComplexMod_gr_221.gif].  

Explore Solution 12.2.

Applying Theorem 12.3 we see that the Fourier series for   [Graphics:../Images/FourierSeriesComplexMod_gr_239.gif]  involves only the cosine terms,  [Graphics:../Images/FourierSeriesComplexMod_gr_240.gif],  and

    [Graphics:../Images/FourierSeriesComplexMod_gr_241.gif].  

We can use Mathematica to construct the coefficients [Graphics:../Images/FourierSeriesComplexMod_gr_242.gif].

[Graphics:../Images/FourierSeriesComplexMod_gr_243.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_244.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_245.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_246.gif]

This is easy to simplify and obtain

    [Graphics:../Images/FourierSeriesComplexMod_gr_247.gif].

 

We can obtain this with the indefinite integral

[Graphics:../Images/FourierSeriesComplexMod_gr_248.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_249.gif]


[Graphics:../Images/FourierSeriesComplexMod_gr_250.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_251.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_252.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_253.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_254.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_255.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_256.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_257.gif]

This is easy to simplify and obtain

    [Graphics:../Images/FourierSeriesComplexMod_gr_258.gif].

 

 

[Graphics:../Images/FourierSeriesComplexMod_gr_259.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_260.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_261.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_262.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_263.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_264.gif]

Thus, we have obtained the desired result

    [Graphics:../Images/FourierSeriesComplexMod_gr_265.gif].

 

Now we continue our explorations.

Enter the function u[t], and for illustration construct the Fourier Series - Trigonometric Polynomial of degree n = 9.

[Graphics:../Images/FourierSeriesComplexMod_gr_266.gif]



[Graphics:../Images/FourierSeriesComplexMod_gr_267.gif]


We can also get this result using Mathematica's built in procedure FourierTrigSeries.

[Graphics:../Images/FourierSeriesComplexMod_gr_268.gif]



[Graphics:../Images/FourierSeriesComplexMod_gr_269.gif]


A graph of  [Graphics:../Images/FourierSeriesComplexMod_gr_270.gif]  is given below.

[Graphics:../Images/FourierSeriesComplexMod_gr_271.gif]



[Graphics:../Images/FourierSeriesComplexMod_gr_272.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_273.gif]


The general term for the series is  [Graphics:../Images/FourierSeriesComplexMod_gr_274.gif].  We can sum up 5 terms and see that it agrees with  [Graphics:../Images/FourierSeriesComplexMod_gr_275.gif].
Then we can sum the infinite series  [Graphics:../Images/FourierSeriesComplexMod_gr_276.gif]  and see if it agrees with  [Graphics:../Images/FourierSeriesComplexMod_gr_277.gif].  

[Graphics:../Images/FourierSeriesComplexMod_gr_278.gif]
   
   
   
   
   
   
   
   
   
   

 

 

We can plot  T[t]  to see what it looks like.

[Graphics:../Images/FourierSeriesComplexMod_gr_280.gif]




[Graphics:../Images/FourierSeriesComplexMod_gr_281.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_282.gif]

Therefore,  [Graphics:../Images/FourierSeriesComplexMod_gr_283.gif] is the periodic extension of  [Graphics:../Images/FourierSeriesComplexMod_gr_284.gif].

 

 

We can compare this solution with the other one that we stated.

[Graphics:../Images/FourierSeriesComplexMod_gr_285.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_286.gif]

[Graphics:../Images/FourierSeriesComplexMod_gr_287.gif]

Therefore,  [Graphics:../Images/FourierSeriesComplexMod_gr_288.gif] is another form of the the periodic extension of  [Graphics:../Images/FourierSeriesComplexMod_gr_289.gif].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell