Examples of Fourier series
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- Peregrine Waters
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1 Examples of Fourier series Preliminaries In[]:= Below is the definition of a periodic extension of a function defined on L, L. This definition takes a function as a variable. The function has to be inputted as a so called pure function (that is instead of the variable we put # and the formula ends with &). Clear ff, x, ; In[5]:= fft ff_, x_, _ : ff x fft 2 &, x, x Ceiling 2 2 Out[5]= x 2 Ceiling x 2 In[7]:= Plot fft 2 &, x,, x, 4, Out[7]=
2 2 Examples_Fourier_series.nb Example - In[3]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ Sign x ; nn 0; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 Out[6]= Out[7]= 0 Out[8]= 0 ca n_, _ FullSimplify Integrate ff x Cos x, x,,, And 0, n Integers, n 0 2 n n Π
3 Examples_Fourier_series.nb 3 In[9]:= ; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4, 7,.5,.5 ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range 0, 4,, PlotRange 4, 7,.5,.5, AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point,, Point,, Point, 0 & Range 0, 3,, PointSize 0.04, White, Point,, Point, & Range 0, 3, ; pic3 Plot Evaluate ca0 Sum ca n, Cos Sum cb n, Sin x, n,, nn, x, 2, 4, x, n,, nn PlotStyle Thickness 0.003, Black, PlotRange 4, 7,.5,.5 ; Show pic, pic2, pic2a, pic Out[9]=
4 4 Examples_Fourier_series.nb Example 0 In[20]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ UnitStep x ; nn 0; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 Out[23]= Out[24]= ca n_, _ FullSimplify Integrate ff x Cos x, x,,, And 0, n Integers, n 0 n n Π 2 Out[25]= 0
5 Examples_Fourier_series.nb 5 In[26]:= ; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4, 7,.5,.5 ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range 0, 4,, PlotRange 4, 7,.5,.5, AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point, 0, Point,, Point, 2 & Range 0, 3,, PointSize 0.04, White, Point, 0, Point, & Range 0, 3, ; pic3 Plot Evaluate ca0 Sum ca n, Cos Sum cb n, Sin x, n,, nn, x, 2, 4, x, n,, nn PlotStyle Thickness 0.003, Black, PlotRange 4, 7,.5,.5 ; Show pic, pic2, pic2a, pic3.5.0 Out[26]= Example In[27]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ x; nn 20; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 Out[30]= 2 n n Π
6 6 Examples_Fourier_series.nb In[3]:= 2; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4,, 3, 3 ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range 0, 4, 4, PlotRange 4,, 3, 3, AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point, 2, Point, 2, Point, 0 & Range 0, 3, 4, PointSize 0.04, White, Point, 2, Point, 2 & Range 0, 3, 4 ; pic3 Plot Evaluate Sum cb n, Sin x, n,, nn, x, 2, 4, PlotStyle Thickness 0.003, Black, PlotRange 4,, 3, 3 ; Show pic, pic2, pic2a, pic3 2 Out[3]=
7 Examples_Fourier_series.nb 7 Example 2 In[32]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ Abs x ; nn 0; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 ca n_, _ FullSimplify Out[35]= 0 Integrate ff x Cos x, x,,, And 0, n Integers, n 0 Out[36]= Out[37]= 2 2 n n 2 Π 2
8 8 Examples_Fourier_series.nb In[38]:= 2; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4,,, 3 ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range 0, 4, 4, PlotRange 4,,, 3, AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point, 2, Point, 2, Point, 0 & Range 0, 3, 4, PointSize 0.04, White, Point, 2, Point, 2 & Range 0, 3, 4 ; pic3 Plot Evaluate ca0 Sum ca n, Cos x, n,, nn Sum cb n, Sin x, n,, nn, x, 2, 4, PlotStyle Thickness 0.003, Black, PlotRange 4,,, 3 ; Show pic, pic2, pic3 2 Out[38]=
9 Examples_Fourier_series.nb 9 Example 3 In[39]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ x UnitStep x ; nn 0; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 ca n_, _ FullSimplify Out[42]= Out[43]= Out[44]= Integrate ff x Cos x, x,,, And 0, n Integers, n 0 n n Π 4 n n 2 Π 2
10 0 Examples_Fourier_series.nb In[45]:= 2; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4,,, 3 ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range 0, 4, 4, PlotRange 4,,, 3, AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point, 0, Point, 2, Point, & Range 0, 3, 4, PointSize 0.04, White, Point, 0, Point, 2 & Range 0, 3, 4 ; pic3 Plot Evaluate ca0 Sum ca n, Cos Sum cb n, Sin x, n,, nn, x, 2, 4, x, n,, nn PlotStyle Thickness 0.003, Black, PlotRange 4,,, 3 ; Show pic, pic2, pic2a, pic3 3 2 Out[45]=
11 Examples_Fourier_series.nb Example 4 In[46]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ x 2 UnitStep x ; nn 20; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 ca n_, _ FullSimplify Out[49]= Out[50]= Out[5]= Integrate ff x Cos x, x,,, And 0, n Integers, n 0 n 3 Π n 2 n 2 Π n 2 n 2 Π 2
12 2 Examples_Fourier_series.nb In[52]:= ; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4, 7,.,. ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range, 4, 2, PlotRange 4, 7,.,., AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point, 0, Point,, Point, 2 & Range, 3, 2, PointSize 0.04, White, Point, 0, Point, & Range, 3, 2 ; pic3 Plot Evaluate ca0 Sum ca n, Cos Sum cb n, Sin x, n,, nn, x, 2, 4, x, n,, nn PlotStyle Thickness 0.003, Black, PlotRange 4, 7,.,. ; Show pic, pic2, pic2a, pic Out[52]=
13 Examples_Fourier_series.nb 3 Example 5 In[53]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ x 2 ; nn 0; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 ca n_, _ FullSimplify Out[56]= 0 Integrate ff x Cos x, x,,, And 0, n Integers, n 0 Out[57]= Out[58]= n 2 n 2 Π 2
14 4 Examples_Fourier_series.nb In[59]:= ; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4, 7,.,. ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range, 4, 2, PlotRange 4, 7,.,., AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point, 2, Point, 2, Point, 0 & Range 0, 3, 4, PointSize 0.04, White, Point, 2, Point, 2 & Range 0, 3, 4 ; pic3 Plot Evaluate ca0 Sum ca n, Cos x, n,, nn Sum cb n, Sin x, n,, nn, x, 2, 4, PlotStyle Thickness 0.003, Black, PlotRange 4, 7,.,. ; Show pic, pic2, pic Out[59]=
15 Examples_Fourier_series.nb 5 Example 6 In[60]:= Clear ca0, ca, cb, ff, n,, nn ; ff x_ x 2 Sign x ; nn 0; cb n_, _ FullSimplify Integrate ff x Sin x, x,,, And 0, n Integers, n 0 ca0 FullSimplify Integrate ff x, x,,, And 0 2 Out[63]= Out[64]= 0 Out[65]= 0 ca n_, _ FullSimplify Integrate ff x Cos x, x,,, And 0, n Integers, n 0 n 3 Π n 2 n 2 Π 2
16 6 Examples_Fourier_series.nb In[66]:= ; Module pic, pic2, pic2a, pic3, pic Plot ff x, x,,, PlotStyle Thickness 0.0, Blue, PlotRange 4, 7,.,. ; pic2 Plot fft ff &, x,, x, 5, 0, PlotStyle Thickness 0.005, Green, Exclusions Range, 4, 2, PlotRange 4, 7,.,., AspectRatio Automatic ; pic2a Graphics PointSize 0.02, Green, Point,, Point,, Point, 0 & Range, 3, 2, PointSize 0.04, White, Point,, Point, & Range, 3, 2 ; pic3 Plot Evaluate ca0 Sum ca n, Cos Sum cb n, Sin x, n,, nn, x, 2, 4, x, n,, nn PlotStyle Thickness 0.003, Black, PlotRange 4, 7,.,. ; Show pic, pic2, pic2a, pic Out[66]=
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