Can the Definite Integral of sinc(x) Be Solved Using Fourier Techniques?

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SUMMARY

The definite integral of sinc(x), specifically sin(ax)/x over the interval [-π, π], cannot be solved using elementary functions. Instead, the result is expressed in terms of the sine integral function Si, as confirmed by Wolfram Alpha and Mathematica. This indicates that while Fourier techniques can provide insights, they do not yield a straightforward solution for this integral. Therefore, for practical purposes, one should rely on the sine integral function for evaluation.

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Chen
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I'm studying a course in Fourier. In a multi-choice question, one of the answers asks for the value of the definite integral of sin(ax)/x over [-pi,pi]. I am wondering if there is a way to calculate this integral (I guess using Fourier techniques) or not.
It is possible that it can't be solved, and the question can be answered by verifying that one of the OTHER answers IS correct, but I just want to make sure I'm not missing something.

Thanks,
Chen
 
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Well, Wolfram Alpha / Mathematica gives the result in terms of the "sine integral" function Si.

So if you're looking for an elementary solution, it is highly likely that it does not exist.
 

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