I've read a couple of sources on Fourier transform. All of them give different coefficients and integration limits. Wikipedia: 1, -infinity, +infinity. Russian Wikipedia: 1/sqrt(2*pi), -infinity, +infinity. MathCad (cfft function): 1/sqrt(4*pi), n/a (cfft gets a list of values).
What do they depend on? units of measurement? conventions? How does the choice of them determine coefficients and limits in inverse Forier transform (though this question is not too important for me now)?
In my case i'm trying to apply Fourier transform to a correlation function of some road microprofile to get its spectrum density. Correlation function is represented as the list of values. I've already got some results in MathCad but still struggling to choose the right coefficient. Coefficients in books on the subject of microprofile differ too. Some book by Тарасик: 2/pi, 0, +infinity. Some book by Молибошко: 2, -infinity, +infinity.
Units of measurement of my correlation function R(x) are sm^2(m), all are real values, x>0. I'm not going to do any calculation on the spectrum density, maybe just find an approximation for it. Which coefficients should i choose? What units will my result be in?
In response to your last comment, let me address each question individually.
1) Yes, $n$ is the number of spatial dimensions. Although you didn't ask about this, when undertaking a multidimensional Fourier transform, we take the inner product (as we are in $\mathbb{R^{n}}$ this is equivalent to the dot product) of the variables being mapped between.
For example, in $1$ dimension we have
$$ \hat f(\omega) = \int_{\mathbb{R}} f(x)\exp(-i\omega x)dx $$
where as if we have $n=2,3,..,N$ dimensions we have
$$ \hat f(\omega) = \int_{\mathbb{R^{n}}} f(x)\exp(-i\omega\cdot x)dx $$
Here, I have omitted the scaling factors in the transform because I don't know the form of the Fourier transform you are used to.
2) I wouldn't say you're "free" to distribute this $\frac{1}{\sqrt{2\pi}}$ however you please.. You are better off sticking with one of the three conventions.
3) When I said in the comments that "Usually if you see a change in the integration limits from
$$\int_{-\infty}^{\infty} f(x)dx$$ to
$$2\int_{0}^{\infty} f(x)dx$$
it is because the function you are integrating is even", I was trying to explain why the author of your book might have done what he/she did. I didn't really understand your question (I still don't), which is why I was hoping you might add a bit more detail to your original post so that I might help you.