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I am trying to formalize the logical steps to prove that the Fourier Series of a function withperiod[itex]\rightarrow \infty[/itex] leads to the Fourier transform. So let's have the Fourier series:

[tex]f(x)=\sum_{n=-\infty}^{+\infty}c_n e^{i\cdot \frac{2\pi n}{L}x}[/tex]

whereLis the period of the functionf.

Many texts simply say that when L tends to infinity, c_{n}becomes a continuous function [itex]c(n)[/itex] and the summation becomes an integral.

[tex]f(x) = \int_{-\infty}^{+\infty}c(k) e^{i\cdot k x}dk[/tex]

Unfortunately they do not explain why, and they do not mention what is the logical step that allows one to switch from the discrete c_{n}to the continuous c(k), and from the summation to an integral withdk.

Any hint?

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# From Fourier Series to Fourier Transforms

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