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From: glen herrmannsfeldt on 30 Apr 2010 15:36
fisico32 <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote:

(snip on FT of integrals)

> thanks for your input. You are right, we get just a number if the limits
> are finite numbers or both infinity, + and -.
> I guess I am considering the case where the two limits are finite numbers
> but still variables.

If those variables depend on t, then you can do a change of variables
and form the sum of two integrals with upper limit t.

> So the final result I am looking for is an expression for the FT of a
> integral of a function with upper and lower limit of integration that are
> real (or complex)variables.....

If they don't depend on t, then you can more the integral out of
the transform. Otherwise, if both depend on t, then it is the
sum of difference of two integrals, each with one limit depending
on t.

> The common result in the book has the upper limit of integration equal to t
> and the lower limit equal to -infinity. I guess that is because we would
> eventually get a 1-dimensional function out of that integral say g(t), to
> be then Fourier transformed.

> If the two limits of integration were both variables we could only do a
> partial fourier transform or need to do a double Fourier
> transform...correct?

I don't know what you mean by double, but usually that would mean
a 2D transform. In this case, it would be the sum or difference
of two transforms.

The Fourier transform is a linear operator.

-- glen
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