Exploration.

Set up the formulas for the piecewise function  [Graphics:../Images/FourierSeriesComplexMod_gr_48.gif].  

[Graphics:../Images/FourierSeriesComplexMod_gr_49.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_50.gif]


               [Graphics:../Images/FourierSeriesComplexMod_gr_51.gif]

  

                              The  left-hand and right-hand limits at  [Graphics:../Images/FourierSeriesComplexMod_gr_52.gif],  [Graphics:../Images/FourierSeriesComplexMod_gr_53.gif],  and  [Graphics:../Images/FourierSeriesComplexMod_gr_54.gif]  are easily determined:

 

        At  [Graphics:../Images/FourierSeriesComplexMod_gr_55.gif],  the left-hand limit is  

                    [Graphics:../Images/FourierSeriesComplexMod_gr_56.gif],     
                    
and the right-hand limit is  
                    
                    [Graphics:../Images/FourierSeriesComplexMod_gr_57.gif].

This computation can be done with Mathematica.

[Graphics:../Images/FourierSeriesComplexMod_gr_58.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_59.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_60.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_61.gif]

The Maple commands are similar  

[Graphics:../Images/FourierSeriesComplexMod_gr_62.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_63.gif]  


[Graphics:../Images/FourierSeriesComplexMod_gr_64.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_65.gif]  

 

        At  [Graphics:../Images/FourierSeriesComplexMod_gr_66.gif],  the left-hand limit is  

                    [Graphics:../Images/FourierSeriesComplexMod_gr_67.gif][Graphics:../Images/FourierSeriesComplexMod_gr_68.gif],

and the right-hand limit is  

                    [Graphics:../Images/FourierSeriesComplexMod_gr_69.gif].

This computation can be done with Mathematica.

[Graphics:../Images/FourierSeriesComplexMod_gr_70.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_71.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_72.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_73.gif]

The Maple commands are similar  

[Graphics:../Images/FourierSeriesComplexMod_gr_74.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_75.gif]  


[Graphics:../Images/FourierSeriesComplexMod_gr_76.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_77.gif]  

 

        At  [Graphics:../Images/FourierSeriesComplexMod_gr_78.gif],  the left-hand limit is  

                    [Graphics:../Images/FourierSeriesComplexMod_gr_79.gif][Graphics:../Images/FourierSeriesComplexMod_gr_80.gif],

and the right-hand limit is  

                    [Graphics:../Images/FourierSeriesComplexMod_gr_81.gif].

This computation can be done with Mathematica.

[Graphics:../Images/FourierSeriesComplexMod_gr_82.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_83.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_84.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_85.gif]

The Maple commands are similar  

[Graphics:../Images/FourierSeriesComplexMod_gr_86.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_87.gif]  


[Graphics:../Images/FourierSeriesComplexMod_gr_88.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_89.gif]  

 

Remark.  For this example, one sided limits will exist at the endpoints  [Graphics:../Images/FourierSeriesComplexMod_gr_90.gif].  

 

        At  [Graphics:../Images/FourierSeriesComplexMod_gr_91.gif],  the right-hand limit is  

                    [Graphics:../Images/FourierSeriesComplexMod_gr_92.gif].

This computation can be done with Mathematica.

[Graphics:../Images/FourierSeriesComplexMod_gr_93.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_94.gif]

The Maple command is similar  

[Graphics:../Images/FourierSeriesComplexMod_gr_95.gif]

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_96.gif]  

 

        At  [Graphics:../Images/FourierSeriesComplexMod_gr_97.gif],  the left-hand limit is   

                    [Graphics:../Images/FourierSeriesComplexMod_gr_98.gif].

This computation can be done with Mathematica.

[Graphics:../Images/FourierSeriesComplexMod_gr_99.gif]
[Graphics:../Images/FourierSeriesComplexMod_gr_100.gif]

The Maple command is similar  

[Graphics:../Images/FourierSeriesComplexMod_gr_101.gif]  

                                                            [Graphics:../Images/FourierSeriesComplexMod_gr_102.gif]  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2010 John H. Mathews, Russell W. Howell