Understanding Fourier Series: Complex vs. Ordinary Coefficients

[zezima1]

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?

 

[HallsofIvy]

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

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

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