FOURIER SERIES
This slide show consists of graphs of various functions together with some of
their Fourier Series approximations.
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A. Triangular Wave
The function here is
x if -ã/2 < x < ã/2
ã - x if ã/2 < x < 3ã/2
and has period 2ã.
The first slide shows this function for -2ã < x < 2ã. The next 6 slides show
the Fourier series expansion out to terms 1, 2, 3, 4, 5, 6, superimposed on the
function.
B. Square Wave
The function here is
-1 if -2 < x < 0
1 if 0 < x < 2
and has period 4.
The first slide shows this function for -4 < x < 4. The next 11 slides show the
Fourier series expansion out to terms 1, 2, 3, ... , 9, 10, 11, superimposed on
the function. Notice Gibbs phenomena.
C. Saw Tooth Wave
The function here is
x if -1 < x < 1
and has period 2.
The first slide shows this function for -3 < x < 3. The next 11 slides show the
Fourier series expansion out to terms 1, 2, 3, ... , 9, 10, 11, superimposed on
the function. Again notice Gibbs phenomena.
D. Cosine Expansion of Sine
The function is
sin x if 0 < x < ã.
The first slide shows this function. The next 8 slides show the Fourier cosine
series expansion out to terms 1, 2, 3, 4, 5, 6, 7, 8 superimposed on the
function.
E. Interrupted Square Wave
The function here is
0 if -ã < x < -ã/2
-ã/2 if -ã/2 < x < 0
ã/2 if 0 < x < ã/2
0 if ã/2 < x < ã
and has period 2ã.
The first slide shows this function for -ã < x < ã. The next 13 slides show the
Fourier series expansion out to terms 1, 2, 3, ... 11, 12, 13, superimposed on
the function.
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