Re: integral of sin(x)/x


>>>The reason that this derivation begs the question is that
>>>you actually have to know that the integral of sin(k)/k
>>>is pi/2 to prove that taking the Fourier transform of
>>>the Fourier transform gives the original function back.
>>
>>No, you don't need this. Some proofs of the fact *use* this,
>>but there are other proofs that don't.

>Well, I don't know of how you would prove it without
>doing something at least as difficult.

Another way of doing this is to use f(x)=exp(-x^2), which
satisfies f'(x)+xf(x)=0. The Fourier transform satisfies
the same differential equation. This trick is used in Rudin's
book on Functional analysis to anchor the inversion theorem.
This is much easier than using sin(x)/x, IMHO.

--Dan Grubb
.

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