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Fourier Transform : Equivalent Width

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"Equivalent Widths"
Suppose we define for a square-integrable function f(t) and its Fourier transform
.....
the equivalent width as
....
and the equivalent Fourier width as

....
a) Show that
.....
is independent of the function f, and determine the value of this const.
b) Determine the equivalent width and the equivalent Fourier width for the unnormalized Gaussian
.....
and compare them with its full width as defined by its inflection points.

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Equivalent width in a Fourier Transform is investigated. The solution is detailed and well presented.

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