Proving Fourier Transform

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  • Thread starter henry wang
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fourierSeries.png

Quote: "The Fourier transform is a generalization of the complexFourier series in the limit as [PLAIN]http://mathworld.wolfram.com/images/equations/FourierTransform/Inline1.gif. [Broken] Replace the discrete http://mathworld.wolfram.com/images/equations/FourierTransform/Inline2.gif with the continuous
Inline3.gif
while letting [PLAIN]http://mathworld.wolfram.com/images/equations/FourierTransform/Inline4.gif. [Broken] Then change the sum to an integral, and the equations become
Inline5.gif
Inline6.gif
Inline7.gif

(1)
Inline8.gif
Inline9.gif
Inline10.gif

"
From: http://mathworld.wolfram.com/FourierTransform.html
Why can we replace A(n) with F(k)dk as L tends to infinity? I know that A(n) will be continues as L tends to infinity, but I cant make sense of F(K)dk.
Can I just let A(n)=F(k)dk or can I proof this from the definition of Fourier series rigorously?
 
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  • #2
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Thx I solved it.
 

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