I made a mistake. The Fourier transform of 1/x actually has a sign function in it:
http://www.wolframalpha.com/input/?i=fourier+1%2Fx
because depending on whether k in the exponential is positive or negative, you have to complete your contour in the upper or lower half respectively of the...
Well if the Fourier transform of 1/x exists, it would make the most sense that it's the principal part.
However, can you really ignore what's going on at x=0 like that?
For example, can you reconstruct the function from the transform?
f(x)=\int \frac{dk}{2 \pi} (i \pi) e^{-ikx}=(i...
Does it make sense to take the Fourier transform of a function that blows up at some point? For example the Fourier transform of f(x)=1/x, which blows up at zero?
Doesn't the integral:
\int^{\infty}_{-\infty} \frac{dx}{x} e^{-ikx}
not converge because of x=0?
Yet for some reason analytical...
Suppose a function f(k) has a power series expansion:
f(k)=\Sigma a_i k^i
Is it possible to inverse Fourier transform any such function?
For example:
f(k)=\Sigma a_i k^{i+2}\frac{1}{k^2}
Since g(k)=1/k^2 should have a well-defined inverse Fourier transform, and the inverse Fourier...
I figured it out. The method I outlined is correct.
So we had this:
\frac{i^{d+2}}{2\pi} \int dk \mbox{ } e^{ik} \left(\frac{1}{k}\right)^{d+2}\left(e^{-ik}-1 \right)^{d+2}
Manipulating:
\frac{2^{d+1}}{\pi} \int dk...
I tried to calculate the volume of a simplex, but got an integral I couldn't do.
For simplicity take a 2-simplex (the volume of a 2-simplex is 1/6)
V=\int da \int dx \int dy \int dz \mbox{ } \delta(1-a-x-y-z)
where the integration limits are over the 4-cube.
My reasoning for this formula...
The integral of ln(1-x) is -(1-x)ln(1-x)-x, when 0<=x<=1.
So for example:
\int_{0}^{1} ln (1-x)dx= (-(1-x)ln(1-x)+x)_{0}^{1}=-1
However, going to an online integrating site:
http://integrals.wolfram.com/index.jsp?expr=Log[1-x]&random=false
they give the integral of ln(1-x) as...
If your measure space is Borel then I don't think this statement is true. A sequence of Borel measurable functions that has a limit converges to a Borel function.