# How to get fourier transform from fourier series

by Ahmed Abdullah
Tags: fourier, series, transform
 P: 183 I think I know the answer now. " For a function periodic in [-L/2,L/2], Fourier series is $$f(x) = \sum_{n=-\infty}^{\infty }A_{n}e^{i(2\pi nx{/}L)} \\A_{n} = 1{/}L \int_{-L{/}2}^{L{/}2}f(x)e^{-i(2\pi nx{/}L)}dx.$$" The part that was bothering me was " The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete $$A_{n}$$ with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become $$f(x) = \int_{-\infty}^{\infty} F(k)e^{2\pi ikx}dk \\ F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx}dx$$ " My question was how $$A_{n}$$ becomes $$F(k) d(k)$$. Especially where the dk comes from? $$k=n{/}L$$; then $$\Delta k = (n+1){/}L -n{/}L =1{/}L$$ , when $$L$$ goes to infinity $$\Delta k$$becomes dk. So when $$L \rightarrow \infty; A_{n} =F(k)dk$$. Indeed! I am a happy man now :).