This is somewhat similar to this question, except the problem I am encountering is to do with Fourier transforms of scalar multiples of functions and their derivatives.
I wish to input
FourierTransform[a*f[t],t,x] and have Mathematica simplify it to
a*FourierTransform[f[t],t,x], and, equivalently, to input
FourierTransform[f'[t],t,x] and have Mathematica simplify it to
ix*FourierTransform[f[t],t,x]. I'm taking the Fourier transform of a system of differential equations for functions
g[t], etc., in order to instead only have to solve a system of algebraic equations for their Fourier transforms, but Mathematica seems to be having some problems doing this.
LaplaceTransform works exactly as expected, but for some reason
FourierTransform doesn't perform the expected simplification. If someone could suggest a solution that also incorporates the distributive property that
FourierTransform was shown to have a problem with in the question I linked, that would be ideal.
This is essentially what I'd like
FourierTransform to do:
FourierTransform of the same expression returns:
There is a solution to this problem here, although it doesn't actually explain
FourierTransform's strange functionality.