Complex fourier series

  • Thread starter zezima1
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The fourier series can also be written as:

f(x) = Ʃcr*exp(r*2π*i*x/L) where sum if from -∞ to ∞

My book says this at least, but I can't really determine the realitionship between the coefficients of an ordinary fourier and the complex one. How do you get rid of the i that would appear in front of every sin factor, and how do you overall translate the coefficients cr to ar and br of an ordinary fourier series?
 

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HallsofIvy
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Use Cauchy's formula:
[itex]e^{2ir\pi x/L}= cos(2r\pi x/L)+ i sin(2r\pi x/L)[/itex]

Cosine is an even function and sine is an odd function so
[itex]e^{-\pi x/L}= cos(2r\pi x/L)- i sin(2r\pi x/L)[/itex]
which is why you do not need negative values of r in the sine, cosine series.

[tex]\sum_{r=-\infty}^\infty a_re^{2ir\pi x/L}= (a_r+ a_{-r}) cos(2r\pi x/L)+ (a_r- a_{-r})i sin(2r\pi x/L)[/tex]
 

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