The Fourier series can easily be represented in complex form, which I examine in this post. To begin with, we consider representing the Fourier trig function in complex form, e.g.
cos npx/ L = [exp inpx/ L + exp - inpx/ L] / 2
sin npx/ L = [exp inpx/ L - exp - inpx/ L] / 2i
And one complex form for the Fourier series is:
f(x) = å ¥-¥ c n exp i2px/ L
And the Fourier coefficient can be written:
c n = 1/2 (a n - i b n ) For n > 0
c n = 1/2 (a n + i b n ) For n < 0
c n = a o / 2
The basic complex Fourier series integral can then be written:
ò -L +L exp inpx/ L exp - impx/ L dx = ò -L +L exp i(np -mp)x/ L dx
And the functions are orthogonal on the interval (-L < x < L) and we obtain for the Fourier coefficient:
c n = 1/2L ò -L +L f(x) exp (- inpx/ L ) dx
Problems for the math whiz:
1) For the function:
f(t) = {0 for h < t < 2p
= {1 for 0 < t < h
Over 0 < t < 2 p h > 0
Give a simplified expression for the associated Fourier coefficient if: c n = 1/2 p ò 2p 0 f(t) exp (-int) dt
= 1/ 2p ò 2p h e - int dt
2) Let k = k1 + ik2 be a complex variable. Let f(x) be a real-valued function of a real variable x and define:
F + (k) = Ö (2/ p ) ò ¥ 0 f(x) exp (ikx) dx
Find F + (k) for: x N e -x
(N an integer)
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