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Instrument Function -- from Wolfram MathWorld

The finite Fourier cosine transform of an apodization function, also known as an apparatus function. The instrument function corresponding to a given apodization function is

(1)

which, upon expanding the complex exponential,

For an even function, the left integrand is even (and hence is equal to twice its value over half its interval) and the right integrand is odd (and hence equal to 0), so

(4)

,

Weisstein, Eric W. "Instrument Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InstrumentFunction.html

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