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A general expression for c
n
The function FOURIER can provide a general expression for the coefficient c
n
of the complex Fourier series expansion. For example, using the same
function g(t) as before, the general term c
n
is given by (figures show normal
font and small font displays):
The general expression turns out to be, after simplifying the previous result,
π
π
π
πππ
in
in
n
en
inniein
c
233
2222
2
232)2(
⋅
−++⋅+
=
We can simplify this expression even further by using Euler’s formula for
complex numbers, namely, e
2in
π
= cos(2nπ) + i⋅sin(2nπ) = 1 + i⋅0 = 1, since
cos(2nπ) = 1, and sin(2nπ) = 0, for n integer.
Using the calculator you can simplify the expression in the equation writer
(‚O) by replacing e
2in
π
= 1. The figure shows the expression after
simplification:
The result is c
n
= (i⋅n⋅π+2)/(n
2
⋅π
2
).
Putting together the complex Fourier series
Having determined the general expression for c
n
, we can put together a finite
complex Fourier series by using the summation function (Σ) in the calculator as
follows:
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