# Fourier Series to Fourier Integrals

I fairly understand the heuristic approach to get to equation 5 in picture 2 from the fourier series in picture 1. Can the explanation be made a little more rigorous? Thank you.

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I've never seen a rigorous development of Fourier integral as a limit of Fourier series. (This does not necessarily mean one does not exist). The books I've read develop Fourier analysis on the circle and on the line separately: An introduction to harmonic analysis by Katznelson is one such book. – user31373 Jun 16 '12 at 17:10