Depending on the definitions, the constant prefactors in continuous and discrete fourier transforms can differ arbitrarily as long as the forward and backward counterparts multiply up to 1/T (period) or 1/2$\pi$ (fourier series/fourier transform). But most textbooks seem inconsistent in defining them for continuous and discrete fourier transforms, in that for instance for continuous 1/2$\pi$ is with the frequency amplitude and for discrete 1/N (# of sampling points) is with the temporal amplitude. Moreover, dimensionality or scaling balance is broken in all definitions as far as I know. How to understand all these and is there a good text with consistency?
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